If the egg is not necessarily of any specific type:
Then it could be said that the egg came first, because other animals had been laying eggs long before chickens existed, such as the dinosaurs. In biology, egg is used as a general term in this way.
If only an egg that will hatch into a chicken can be considered a chicken egg: Then a re-consideration of the original question suggests: Some animal other than a chicken laid the first chicken egg which contained the first chicken. In this case the chicken egg came before the chicken.
In reality, many scientific theories suggest that this would not have been a simple event. For example, the theory of Punctuated equilibrium theorizes that the actual speciation of an organism from its ancestral species is usually the result of many mutations combined with new geographical surroundings, called Cladogenesis.
If only an egg laid by a chicken can be considered a chicken egg: Then a re-consideration of the original question suggests: The first chicken (which hatched from a non-chicken egg) laid the first chicken egg. In this case the chicken came before the chicken egg. Again, this would not necessarily be a straightforward event.
If only an egg that is laid by a chicken and that will hatch into a chicken can be considered a chicken egg: Then the first chicken came from a different type of egg (not a chicken egg) and laid the first chicken egg. In this case eggs (in general) came first, the chicken came after, and the chicken egg came last.
so you decide