If you hold an egg that hatches a chicken, would you call this a chicken-egg? Most definitely.
The only possible way for the chicken to have come before the egg is if you deny this definition of a chicken-egg. Proof:
Assume towards a contradiction that the chicken came first. That chicken hatched from an egg (as we know eggs in general predates chickens). Thus, that chicken hatched from a chicken-egg by the definition above. Therefore, we have a contradiction, as that egg came before the chicken.
The egg came first!