No one questions the date of Cyrus' conquest of Babylon: 539 b.c.e., nor his release of the Jews in 537 or 538.

Four different Bible writers say the Jews were in captivity 70 years:

• Jeremiah 25:11;
• Daniel 9:2;
• 2 Chronicles 36:21;
• Zechariah 1:12.

70 years before 537/38 would be 607/608. The 'scholars' who are stuck on 587 or 586 are relying on a Latin copy of an Armenian script translated from a lost Greek record written by Eusebius who quotes a timeline of Babylonian kings written down 200 years after the fact by a Babylonian priest named Berossus (the same Berossus who wrote about dog-headed horses and fish who walked upright and spoke).

You can read more about biblical derived chronology here.

7 year old question, but still some evidence not considered. Original post brings up the difference between what JWs believe (607 BCE) and what secular sources quote (586/587 BCE).

The return date is pretty undisputed- 537 BCE. And the Bible specifies an exile of 70 years. So, a 70 year exile ending in 537 BCE would have to start in 607 BCE. Secular sources don't mind at all refuting the Bible, so they're OK with a 50 year exile starting in 587... but as many people cite this date as more authoritative than scripture... most people just know "that's the date we're told (sure that someone has some sort of physical evidence)."

The physical evidence is a Babylonian tablet labeled VAT 4956

The opening line of this tablet reads: “Year 37 of Nebukadnezar, king of Babylon.” Thereafter, it contains detailed descriptions of the position of the moon and planets in relation to different stars and constellations. Also included is one lunar eclipse. Scholars say that all these positions occurred in 568/567 B.C.E., which would make the 18th year of Nebuchadnezzar II, when he destroyed Jerusalem, 587 B.C.E. But do these astronomical references irrefutably point only to the year 568/567 B.C.E.?

The tablet mentions a lunar eclipse that was calculated as occurring on the 15th day of the third Babylonian month, Simanu. It is a fact that a lunar eclipse occurred on July 4 (Julian calendar) of this month during 568 B.C.E. However, there was also an eclipse 20 years earlier, on July 15, 588 B.C.E.

If 588 B.C.E. marked the 37th year of Nebuchadnezzar II, then his 18th year would be 607 B.C.E.—the very year indicated by the Bible’s chronology for the destruction of Jerusalem! (See the time line below.) But does VAT 4956 provide further corroborating evidence for the year 607 B.C.E.?

In addition to the aforementioned eclipse, there are 13 sets of lunar observations on the tablet and 15 planetary observations. These describe the position of the moon or planets in relation to certain stars or constellations.18 There are also eight time intervals between the risings and settings of the sun and the moon.

Because of the superior reliability of the lunar positions, researchers have carefully analyzed these 13 sets of lunar positions on VAT 4956. They analyzed the data with the aid of a computer program capable of showing the location of celestial bodies on a certain date in the past. What did their analysis reveal? While not all of these sets of lunar positions match the year 568/567 B.C.E., all 13 sets match calculated positions for 20 years earlier, for the year 588/587 B.C.E.

One of the places where the lunar observations fit 588 B.C.E. even better than 568 B.C.E. is shown in the tablet reproduced on these pages. On line 3 of that tablet, we read that the moon was in a certain position on the “night of the 9th [of Nisanu].” However, the scholars who first dated the event to 568 B.C.E. (astronomical -567) acknowledged that in 568 B.C.E., the moon was in that position on “the 8th of Nisanu and not on the 9th.” To support dating the tablet to 568 B.C.E., they postulated that the scribe erroneously wrote “9” instead of “8.” But the lunar position in line 3 finds an exact match on Nisanu 9 of 588 B.C.E.

Clearly, much of the astronomical data in VAT 4956 fits the year 588 B.C.E. as the 37th year of Nebuchadnezzar II. This, therefore, supports the date of 607 B.C.E. for Jerusalem’s destruction—just as the Bible indicates.
When Was Ancient Jerusalem Destroyed?​—Part Two – jw.org

... so... the only evidence that supports 587 BCE... actually confirms 607 BCE.

The strangest outlier come from Jewish sources that put the Babylonian siege starting at 425 BCE, with the temple destroyed in 423 BCE, and the return to israel at 353 BCE. Chabad.org

The issue at hand seems to be how to define the 70 year captivity. If the Jews were to return to rebuild in 537BC, it would make the 587 BC allowing only 50 years of captivity. However, if one take the completion of the temple in 517 BC as the end-point of captivity, then there would be 70 years of captivity. It is a compelling argument that the destruction of the temple and the rebuild of the temple being the beginning and end of the 70 year period. The reasons are: 1)the timing could be established more clearly, as opposed to the return of people in waves; 2) temple is the ultimate symbol for the people.

Could this 'çonquest' in 607 BCE be the 'siege' of Jerusalem in Daniel 1:1, in the third year of King Jehoikim ? It seems to be the same event as that in 2 Kings 24:1 - looking at the sequence of surrounding events. It appears to not be a particularly destructive "seige", although the first Hebrew captives were taken to Babylon. They were select young males, taken not to do construction work but to be instructed and prepared to stand before the King. This all gives a total of three Babylonian sieges of Jerusalem with the temple probably destroyed around 587 BCE. These dates are interesting though. Dave.

What makes life a wee bit tricky(!) is that the Bible sometimes gives the year of the reign (eg "in the eighteenth" of King Big-Wig) using the Jewish method of reckoning and sometime gives it using the Babylonian method of reckoning. The Jewish method was the non-accession year method and the year started on the new moon on the 1st Tishri (about September) and the Babylonian method was accession year with the year starting on the new moon on 1st Nisanu (about Mar/Apr). In both methods the year number of the reign is increased by 1 on New Year's Day, but in Accession Year dating the year of reign becomes 1 after the first New Year's Day (before that it was "the year of his becoming king"/accession year) and in non-accession dating the year of reign becomes 2 after the first New Year's Day, and the 11 months and 17 days or the single day(!) of reign before that was the 1st year.

The salient example of use of both the Jewish and Babylonian method is found in the same chapter, Jeremiah 52:12 ("in the nineteenth year") and Jeremiah 52:29 ("in the eighteenth year"). Now Jeremiah 52:28 to 52:30 appears to be a summary from Babylonian sources because verse 28 speaks of taking captive in Neb's 7th year (cf 2 Kings 24:12, the Jewish dating method) and the Babylonian Chronicle BM 21946, Grayston's ABC5 also speaks of this event happening in Neb's 7th year:

[Rev.11'] In the seventh year, the month of Kislîmu [OP: Hebrew, Kislev, 9th month], the king of Akkad mustered his troops, marched to the Hatti-land,

[Rev.12'] and besieged the city of Judah and on the second day of the month of Addaru [OP: Hebrew Adar, 12th month] he seized the city and captured the king.

[Rev.13'] He appointed there a king of his own choice, received its heavy tribute and sent to Babylon. See https://www.livius.org/sources/content/mesopotamian-chronicles-content/abc-5-jerusalem-chronicle/

From this we can conclude that Jer 52:28-30 is using the Babylonian method and may simply be quoting from Babylonian sources. This means that Jerusalem was burned in the 18th of Neb. Babylonian method. Using "Babylonian Chronology - 626 BC to AD 75" by Richard A. Parker and Waldo Dubberstein (1956), the standard work for chronology for the date range, 10 Ab (5th month) of the 18th of Neb.(Jer 52:12 with year changed from Jewish method "19th" to Bab. method "18th") equates to 28 August 587 BC Julian, 22 August 587 bc Gregorian.

Rodger Young also claims that only 587 BC can satisfy all the data, in his paper "When did Jerusalem fall?" www.rcyoung.org

As for the Jehovah's Witness teaching, the date of the destruction of Jerusalem is critically important for Jehovah's Witnesses, and it is critical it is 607 BC. From that they get the date 1914 AD, which underpins their whole theology. In 1914 AD they claim that Jesus returned invisibly, and a few years after that Jesus declared that only the Jehovah's Witness organisation was acceptable to him and Jehovah God. If 607 BC is wrong, then there would be little to undergird 1914 AD as the date of Jesus return, nor any JW doctrine to support his declaring of the Watchtower Bible and Tract Society as the only religious organisation acceptable to Jehovah God.

Carl Olof Jonsson's "The Gentile Times Reconsidered" (1968) is very thorough. It is an expansion of his investigations into the date of the destruction of Jerusalem, It is all on his website here http://kristenfrihet.se/english/epage.htm

For a more thorough answer using more Biblical information, and examining the JW position in more detail, see my answer at https://christianity.stackexchange.com/questions/78711/evangelical-christians-claim-jerusalem-was-destroyed-in-either-587-or-586-bce-w/78765#78765

According to our present calendar and the most widely accepted chronology the year was 587.

Considering only the date of 607 given by Jehovah's Witnesses: it is quite incompatible with almost everything else we know.

One of the most prominent proponents of this 'theory' is Rolf Furuli, offering us his modestly named 'Oslo-chronology' in his book:

— Rolf Furuli: "Persian Chronology and the Length of the Babylonian Exile of the Jews Assyrian, Babylonian, Egyptian and Persian Chronology Compared with the Chronology of the Bible", R. Furuli: Oslo, 2003)

This work relies quite heavily on many quite 'original' observations, conclusions and cherry picking of evidence. It was reviewed as:

Once again we have an amateur who wants to rewrite scholarship. […] Part of his redating is fairly modest: he accepts the beginning and end of Achaemenid rule according to the standard dating, and puts the beginning of Darius I’s reign only one year later than is conventional. He argues, however, that the first 11 years of Xerxes’ reign overlap with the last 11 of Darius, and that Artaxerxes I came to the throne in 475 BCE and ruled 51 years. (F. has indeed found the interesting fact that a couple of tablets have the years ‘50’ and ‘51’ for Artaxerxes, but he admits that overwhelmingly tablets make 41 his last year and none is found between 41 and 50, suggesting the obvious: a scribal error.) Gifted amateurs have sometimes revolutionized scholarship, notably M. Ventris and Linear B. But Ventris was willing to work with specialists such as J. Chadwick whereas F. shows little evidence of having put his theories to the test with specialists in Mesopotamian astronomy and Persian history. Perhaps the most telling point is his rather naive argument that the 70 years of Judaean captivity must be a literal 70 years of desolation of the land because some biblical passages make such a statement. A second volume is promised; we shall see if it is any more convincing.
_{— Lester L Grabbe: Review of "FURULI, ROLF, Persian Chronology and the Length of the Babylonian Exile of the Jews Assyrian, Babylonian, Egyptian and Persian Chronology Compared with the Chronology of the Bible, 1 (Oslo: R. Furuli A/S [[email protected]], 2003), pp. 251. n.p.", in: Journal for the Study of the Old Testament, 28(5), 40–58,'3. History, Geography and Sociology', 2004. DOI}

An argument that rests entirely on word for word literal truth of every passage in the bible is a very weak one.

While Furuli's picking of 'evidence' is sometimes straightforwardly transparent – like when in chapter three he asserts that while the change of reign from Nebuchadnezzar to Amel-Marduk and Neriglissar is documented in tablet NBC 4897; because this contradicts F.’s chronology, he says it “cannot be used” – his most important argument rests the already mentioned on this page tablet VAT 4956.

And this tablet is egregiously misdated by JW-adherents and Furuli.

A more detailed analysis of some aspects and problems with the book, its methodology is here by Hermann Hunger, Vienna, Austria, author of " Astronomical Diaries and Related Texts from Babylonia, Vol. I: Diaries from 652 B.C. to 262 B.C.", Verlag der österreichischen Akademie der Wissenschaften: Wien, 1988)."

A layman's compatible analysis of the astronomical calculations and implications would read like:

The cosmic fingerprint doesn’t lie. Year 37 was 568 BCE, so Jerusalem was destroyed in Year 18, 587 BCE. Watchtower chronology doesn’t stand a chance.
_{— "The Astronomical Diary, VAT 4956", XJW Friends, 18 Feb 2018.}

[…] three lunar eclipse tablets that establish Nebuchadnezzar’s reign: LBAT 1419, LBAT 1420, LBAT 1421. Even these are not the only ‘game in town’, but they are enough to provide dozens of absolute dates that prove Nebuchadnezzar’s 18th year, the year Jerusalem was destroyed, was 587 BC.
_{— VAT 4956}

And in
— John M. Steele and Annette Imhausen [eds.]: "Under One Sky: Astronomy and Mathematics in the Ancient Near East", Ugarit-Verlag: Münster 2002, pp421–428, we see F. Richard Stephenson and David M. Willis have in their chapter "The earliest Datable Observation of Aurora Borealis" en passant also evaluated the lunar data in VAT 4956 and come to the conclusion that the date 586/7 BC can be “confidently affirmed”.