What does Jehovah's Witnesses canon say about the United States, if anything?

Jehovah's Witnesses do not have any texts besides the Bible that is considered "canon". Also we believe that until God's Kingdom takes over the world, all human political governments, both the good and the bad, end up being part of the same world ruled by God's enemy, Satan. That Satan is the ruler of our world can be seen in his offer to Jesus recorded in Luke 4:5-7

5 So he brought him up and showed him all the kingdoms of the inhabited earth in an instant of time. 6 Then the Devil said to him: "I will give you all this authority and their glory, because it has been handed over to me, and I give it to whomever I wish. 7 If you, therefore, do an act of worship before me, it will all be yours."

(Jesus also referred to Satan as the ruler of our world in John 12:31 and John 14:30.)

We believe that the United States is referenced in some of the Bible prophecies, although not alone but together with the United Kingdom with which it has a special relationship. These two governments form the Anglo-American World Power.

Jehovah's Witnesses believe that the Anglo-American World Power is referred to in the following prophecies.

In the book of Daniel:

• As the last world power in the immense image seen in a dream by King Nebuchadnezzar, as described in the second chapter of Daniel (Daniel 2:31-45). The dream image showed multiple world powers, starting from the head symbolizing ancient Babylon ruled by King Nebuchadnezzar, and ending with the feet symbolizing the last world power. Daniel 2:44 tells us that that last world power would be present at the time during which "the God of heaven will set up a kingdom that...crush and put an end to all these kingdoms".

  The feet representing the last world power were described as made from "iron and clay", referring to the fractured, polarized political state of the Anglo-American World Power at the time of the end.

• As the "king of the south" who would "in the time of the end" (Daniel 11:40) engage in "a pushing" against the "king of the north". The "king of the north" is viewed as referring to Russia as is explained in this video.

• As the horn that grew from the head of a fearsome beast (Daniel 7:7, 8). The fearsome beast symbolized the Roman Empire and its 10 horns symbolize the powers that came out of the empire after its disintegration. This conspicuous, "arrogant" (Daniel 7:8) horn was foretold to grow larger than the other horns (Daniel 7:20) and would "humiliate three kings" (Daniel 7:24). The three kings that the United Kingdom humiliated on its way to its position as world empire were Spain, France, and the Netherlands.

In the book of Revelation:

• As the seventh head of the wild beast in Revelation 13:1. The reason for this is that this wild beast represents a composite of human political nations. Its heads represent the political powers that have had a leading involvement with God's people in some way, as explained here.
• As the "two-horned beast" in Revelation 13:11-15. This is described as having "two horns as the lamb", meaning the Lamb of God (John 1:29) and coming "out of the earth", meaning the more stable elements of human society. One reason why this is viewed as representing the Anglo-American world power is because of its role in setting up the "image of the wild beast" which was first the League of Nations and, after second world war, the United Nations.
• As "the false prophet" in Revelation 19:20. This is a continuation of the role that the "two-horned beast" was described as having when making the inhabitants worship the first wild beast (Revelation 13:12; "worship" here means in the sense of giving political powers something that rightfully only belongs to God as the creator).

The reference in Revelation 19:20 is the last reference in the Bible to the Anglo-American World Power as a functioning entity. A few chapters earlier, in Revelation 16:13, "the false prophet" is described as gathering the kings of the entire inhabited earth to "the war of the great day of God the Almighty", i.e. Armageddon.

Armageddon is the fulfilment of the prophecy in Daniel 2:44, a worldwide situation in which all earthly political entities come to their end.

Reference: see this article.