Thanks to Charles Darwin, we now have an answer to this hoary old chestnut:
Species evolve from other species over time. Because evolution is a mostly slow and gradual process, it is difficult—bordering on impossible—to state categorically when one species actually becomes another. It's a bit like trying to state the precise point at which red becomes orange in a rainbow: different people will have different views.
But one thing is for certain: however you choose to define a chicken, if you go far enough back in time in the avian 'family tree', you will come to the first ever chicken. That first chicken's mother was undoubtedly a bird very similar to, but not actually (according to your chosen definition), a chicken. And this chicken-like bird laid the egg from which the first chicken hatched.
So the egg came before the chicken.