Do annihilationism and universalism argue against eternal life?

Annihilationists and universalists argue that the Greek word translated "eternal" (aionion/aionios) is more appropriately translated "eon-like", and thus New Testament terms like "eternal fire/punishment" (in e.g. Matt 18:8, 25:46) should be interpreted to mean long lasting but still finite fire and punishment. Given that argument, do annihilationists and universalists have any New Testament basis for believing that "eternal life" is really eternal and not merely long lasting?