The egg came first. Evolutionarily, birds evolved from reptiles. The first amniotic eggs evolved about 340 million years ago, the first bird-like animals evolved about 150 million years ago. Chicken-like animals evolved about 58,000 years ago with the first true chicken coming along only about 8,000 years ago.
While eggs clearly came before birds and chickens, what about specifically “chicken eggs?” Which came first, the “chicken animal” or the “chicken egg?” This answer is less clear, but I believe the chicken egg came first. That first true chicken hatched from an egg laid by a bird that was not quite a chicken. In evolution, one trait evolves after another. At some point, a non-chicken animal laid a “chicken egg” with the final trait that defines a chicken. So, that chicken egg hatched the first true chicken.
Defining that last trait or set of traits might be quite difficult and is perhaps a meaningless nuance. After all, after that first true chicken mated with an almost true chicken animal, their offspring were likely a mix of chicken-like and true-chicken animals. Okay, now we’re getting into the weeds a little too much, but I think you get the idea.