Does God being eternal mean he is unchanging? Why?

No - it is possible for things to change over time.

I would also argue that God himself in the Bible undergoes numerous changes - God in Job and Genesis acts as part of a pantheon of gods, and clues to this are evident if you understand other ANE cultures. God is neither omniscient nor is he omnipotent and he is VERY anthropomorphic.

However, by the time of the Exodus and the establishment of the kingdoms, God almost literally becomes the all-seeing eye where David declares "nothing is hidden from your sight". It is around this time that Satan becomes the arch-enemy of God.

By the time we get to the Babylonian conquest and expulsion, God has become so angry that he's commanding prophets to lie on their side for years at a time and bake bread with excrement.

And then the final metamorphosis of God: loving father to Jesus.

So even if you wanted to press the case God is eternal, you can't argue that God never changes.