The early Church Fathers, St. Thomas Aquinas, and many creationists today would tell you the chicken came first, because the Book of Genesis reveals that God created birds on the fifth day, along with sea monsters; and we can presume that these original birds emerged ex nihilo as full-fledged adults, or they wouldn't have been able to "fly above the earth across the dome of the sky" (Gen 1:20). Just as Adam and Eve were never infants -- and accordingly lacked navels -- the first chickens never had to hatch.
The Indian Upanishads have it the other way around.
They declare that a cosmic egg emerged from nonexistence, then split into the earth and sky and birthed the sun.
It's clear that somebody's wrong, but it's not clear who.
Now, evolutionary biologists will tell you another thing altogether. Eggs came first, but not chicken eggs. Egg and sperm cells evolved a billion years ago, once primordial organisms had become too complex to copy themselves asexually. But we all know those eggs don't count. So 300 million years ago, reptiles developed their own kind of external egg, with a leathery skin and an internal food supply. Their descendents the birds came onstage 100 million years ago with their own revisions of reptilian egg-technology.
Of course, these scientists have sidestepped the issue. They always do. The question is more philosophical, or metaphysical: which form produced the other, and to what end? One rich response comes from Samuel Butler, the Victorian writer and critic, who gave a lot of serious thought to the old joke that a chicken is just one egg's way of making another egg -- and he ended up suggesting that the egg lays the hen as much as the hen lays the egg. Those artistic types can be very unconventional.
If you insist on asking this question, you'll just have to come up with your own answer: it certainly doesn't matter much either way.