[ZenGum]

Assuming the truth of evolution, we can reason as follows.

If by "egg" you mean "any egg" rather than "chicken egg" then obviously eggs evolved first (we have fossilized dinosaur eggs).

Somewhere along the evolutionary path from ancestral birds to chickens, must be some point at which an older generation is not considered a chicken but a younger generation is*.
So if by "egg" you mean "chicken egg", I must further ask, by "chicken egg" do you mean "egg which was laid by a chicken" or "egg out of which hatched a chicken".
If the former (laid by), then the chicken came first. If the latter (hatched from), then the egg came first.

I cannot think of any reason to favour the "hatched from" interpretation over the "laid by" interpretation or vice versa. Which kind of leaves me stuck, but stuck amidst clarity.



* This claim would be disputed by a few specialist academic types. Sod 'em.