at some point some proto-chicken ancestor laid an egg that was different enough genetically that it counts as a chicken. In other words, a non-chicken laid a chicken egg

This is incorrect. If I take an ostrich egg, empty it out through a small hole, then put a chicken fetus inside, it does not suddenly become a chicken egg. We must therefore conclude that “chicken egg” can only reasonably be defined as an egg laid by a chicken.

The proto-chicken ancestor can never lay a chicken egg, it can at most lay a proto-chicken egg which by some mutation contains a chicken. Therefore the chicken came first.