If a chicken egg is an egg that hatches into a chicken, then unfertilized chicken eggs would not be chicken eggs. But if you took an alligator egg and transplanted a developing chicken embryo into it, that would become a chicken egg.
You'd get the heuristic "All chickens have hatched from chicken eggs", which sounds pretty elegant.

If a chicken egg is an egg laid by a chicken, then you couldn't reliably say that a chicken egg hatches into a chicken - the heuristic from before would become "Not all chickens have hatched from chicken eggs". And that one, while it feels a bit imprecise, might be closer to what we observe in reality, especially with that Proto-chicken argument. So the Proto-chicken would have laid a Proto-chicken egg, which hatched into a chicken, which laid chicken eggs.
And it would work with the current scientific hijinks like hatching chickens from different eggs or straight from test tubes.