Do annihilationism and universalism argue against eternal life?

Late to the party, but as an annihilationist the short answer is that generally one of two things will be the case:

1 - 'Eternal' refers not to the process but the result, as with 'eternal judgment', 'eternal redemption' etc, as ninthamigo says. In other words just as we are not being redeemed forever, but rather redeemed once and the result (the state of being redeemed [adjective]) lasts forever, so eternal punishment, eternal destruction (which is something of an oxymoron if eternal torment is held) refer to being punished/destroyed once, and the result (the state of being dead) lasts forever.

2 - 'Eternal' doesn't refer to a time period at all, but is qualitative like 'holy' or 'heavenly'. I can't think of any examples off the top of my head; perhaps 'eternal fire', which is said to have been what destroyed Sodom and Gomorrah in Jude 1:7, but those two cities aren't still burning.

I've never heard an annihilationist claim that an instance of 'eternal' refers to a duration other than infinity; either it's forever, or it's not referring to time. Thus there is no contradiction to believing eternal life means life without end, as nothing in the relevant passages indicates it is anything else.

When something is unclear in one verse refer to others and frequently the collective view this gives you makes the correct understanding clear.

Revelation talks about the 2nd death. If one is permanently tortured through eternity there would be no second death. Only annihilation in fire can support this reading. Death itself is cast into the fire last. This is symbolic of death never occurring again. It occurs at that precise point in time, because there is death, permanent and eternal death, right up to that point, but not after.

Note that neither Hebrew nor Greek have a word that distinguishes eternal from long lasting. See Sodom and Gomorrah which were both burnt with “eternal” fire.

See many other verses which talk of the destruction and death of the wicked. Correctly viewed one must conclude that this use of the word [aionion] means a fire which like the fires of Sodom and Gomorrah, burn until their task is accomplished, and that the only eternal part of the fire is the consequences.

The wording in the verse is unclear unless you read all the other verses which are completely clear on what happens before you are able to correctly understand what this verse is telling you.

So no, it does not argue against eternal life just because Greek had no word for eternal that was different from long lasting.

The views expressed below are not my own, but your question merits an answer that I hope represents annihilationism as an annihilationist might. I will seek to demonstrate you are correct in believing some annihilationists resort to alternate translations about hell which present difficulties for the traditional understanding of verses about heaven. Nevertheless, understood consistently, annihilationists can still believe that heaven is an eternal state.

1. Some annihilationists believe in translating references to eternity as 'eons' or 'ages'

Even annihilationists will admit that certain verses in scripture can be difficult to interpret from their perspective. Greg Boyd writes:

The most difficult passages for annihilationists to explain are Revelation 14:10-11 and 20:10. These passages speak of the wicked being tormented “day and night forever and ever.”

The solution seems to be a matter of translation, accompanied, as I will suggest below, with a paradigmatic shift in the way we think about all such references to the 'temporal' nature of hell. Boyd Goes on to suggest that 'forever and ever' doesn't have to be a reference to eternity.

The phrase “forever and ever” can be translated “for ages upon ages” which implies an indefinite, but not necessarily unending, period of time.

This leads the a difficulty regarding verses about heaven. If the same trick is used in verses like Matthew 25:46 to make "eternal punishment" into something less than eternal, it seems to undermine the eternality of "eternal life" mentioned in the exact same verse using the exact same word aiōnios.

2. Annihilationists can still believe in the eternity of the heavenly reward

Although seeming to present a difficulty if that were how Matthew 25:46 is translated that way, annihilationism both offers a new translation and an entirely new way of understanding the point of this verse. Boyd also writes:

Now, Scripture certainly teaches that the wicked are punished eternally, but not that the wicked endure eternal punishment. The wicked suffer “eternal punishment”(Mt 25:46), “eternal judgment” (Heb 6:2) and “eternal destruction” (2 Thess 1:9) the same way the elect experience “eternal redemption” (Heb 5:9, 9:12). The elect do not undergo an eternal process of redemption. Their redemption is “eternal” in the sense that once the elect are redeemed, it is forever. So too, the damned do not undergo an eternal process of punishment or destruction. But once they are punished and destroyed, it is forever. Hell is eternal in consequence, not duration. The wicked are “destroyed forever” (Ps 92:7), but they are not forever being destroyed.

Therefore, the solution for an annihilationist isn't merely to translate words like aiōn as 'ages' but to also suggest the meaning of temporal references to eternity - both heaven and hell - is not in fact primarily temporal but descriptions of finality and intensity. Therefore, while 'complete' destruction implies full and just punishment that doesn't need to be eternal, to be 'completely' redeemed places one entirely outside of the clutches of sin, death, and the devil meaning that even without the use of the word eternal, the redeemed are implicitly redeemed forever because they have been redeemed 'completely'. This certainly softens the temporal references to the lasting eternal state of the redeemed of heaven, but it also seems to do justice to the different connotations a descriptor can take when applied to the different concepts of 'destruction' and 'life'.

I am not an expert in biblical Greek so I suppose an expert could mount a critique of the degree to which this explanation depends on English translations and grammar, but this is the way Boyd, as an annihilationist, explains his perspective.