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.
The question can be interpreted in different ways. If you interpret "chicken" to be literally the Chicken species, then indeed a different species laid an egg that became the first chicken. But if you interpret "chicken" more figuratively, then it came first. The first organisms did not lay eggs, so there was a parent that laid the first egg. Either way, Darwin provides the answer.
I think you have to take such questions literally. If the chicken is figurative, then surely the egg must be too—which gets you nowhere.