Eggs were around long before chickens.
Or do you specifically mean a chicken egg?
At what exact moment in time did the species we call "chicken" come into existence? Presumably there was an egg-laying "pre-chicken" that through many, many generations eventually produced something that would be considered a modern chicken.
While we can't pinpoint the exact moment that the chicken species came into existence (it's an arbitrary distinction anyway), presumably the "pre-chicken" begat the first "chicken" at the moment of speciation, meaning that the genetic material that became the first chicken was in fact first present in an egg laid by the "pre-chicken." So the egg came first.