Here are two passages that could be used to attempt to prove annihilationism:
Psalm 37:20 "But the wicked shall perish, and the enemies of the Lord shall be as the fat of lambs: they shall consume; into smoke shall they consume away." (KJV)
Matthew 10:28 "And fear not them which kill the body, but are not able to kill the soul: but rather fear him which is able to destroy both soul and body in hell." (KJV)
The Psalm says the wicked will consume away into smoke, which if you take literally, and understand to be referring to hell, can support annihilationism. And Jesus says God can "destroy" both body and soul in hell. He doesn't say "torment," although in the traditional view everyone pretty much functions as if he had.
Now, as to the question "How can it be everlasting punishment if the person is not conscious?" If you understand the punishment itself as being the cessation of existence, well, that lasts forever. If you burn up in hell and cease to exist, you cease to exist for all eternity: that's everlasting destruction. That is, you are destroyed with absolute finality. Its a punishment, and its lasts forever. It is not, however, a torment that lasts forever.
The problem here is the confusion of the terms punishment and torment, which do not have to be equivalent. Those who hold the traditional view of hell will have a hard time seeing the point that punishment doesn't have to mean torment. But annihilationists will probably be puzzled that anyone doesn't see the distinction between the two.
In terms of this world, the death penalty is a punishment, even though its torment doesn't last forever, and it is an everlasting punishment, in the sense that once you're dead you stay dead (from the worldly perspective). Annihilationism is the more absolute version of the death penalty. Because when someone is put to death by the state, their soul still exists, but annihilationism maintains that God destroys the wicked both body and soul in hell, so that they completely cease to exist. Which brings us back to Matthew 10:28, which clearly compares physical death which doesn't destroy the soul to hell which it is asserted can.