Obviously the chicken would come first because it would be the source of the egg. Actually there would have to be a first chicken and a first rooster to accomplish the first fertilized egg resulting in a chick. So the answer is, like with everything else, a full formed first chicken came first–created by the great I AM.
Obviously a living animal would be necessary in order to give birth to offspring, so the real question is, when did the first chicken-like bird first appear on the earth?
The theory of evolution states that species change over time via mutation and sexual reproduction. Since DNA (deoxyribonucleic acid) can be modified only before birth, a mutation must have taken place at conception or within an egg such that an animal similar to a chicken, but not a chicken, laid the first chicken eggs. These eggs then hatched into chickens that inbred to produce a living population. Hence, in this light, both the chicken and the structure of its egg evolved simultaneously from birds that, while not of the same exact species, gradually became more and more like present-day chickens over time.
Not any mutation in one individual can be considered as constituting a new species. A speciation event involves the separation of one population from its parent population, so that interbreeding ceases; this is the process whereby domesticated animals are genetically separated from their wild forebears. The whole separated group can then be recognized as a new species.
The modern chicken was believed to have descended from another closely related species of birds, the red junglefowl, but recently discovered genetic evidence suggests that the modern domestic chicken is a hybrid descendant of both the red junglefowl and the grey junglefowl. Assuming the evidence bears out, a hybrid is a compelling scenario that the chicken egg, based on the second definition, came before the chicken.
This implies that the egg existed long before the chicken, but that the chicken egg did not exist until an arbitrary threshold was crossed that differentiates a modern chicken from its ancestors. Since this arbitrary distinction cannot be made until after the egg has hatched, one would have to first find the original chicken, then from this find the first egg it laid.
A simple view is that at whatever point the threshold was crossed and the first chicken was hatched, it had to hatch from an egg. The type of bird that laid that egg, by definition, was on the other side of the threshold and therefore not technically a chicken — it may be viewed as a proto-chicken or ancestral chicken of some sort, from which a genetic variation or mutation occurred that thus resulted in the egg being laid containing the embryo of the first chicken. In this light, de facto, that the argument is settled and the egg had to have come first.