There are three possible ways to answer this.
- In the evolutionary lineage of chickens, there is, of course, a continuum of forms of chickens and chicken egg, ranging seamlessly from non-chickens to chickens. Where you conceptually draw the line between “chicken” and “non-chicken” or “chicken egg and “non-chicken egg” in that lineage is obviously arbitrary, but wherever you draw it, if you draw the line on the basis of DNA patterns (as you probably should, as DNA is the blueprint of an organism, determining its every aspect, and it offers a convenient, linear and discrete way to make objective lineal differentiations down to the most minute detail), the first entity to have that DNA pattern would have to be an egg, not a chicken, since the DNA only mutates and recombines from one generation to the next during the creation of the zygote, or egg.
Or you could say that it’s impossible, or ad hoc/arbitrary at best, to make such a precise definition of chicken or chicken egg that you can delineate exactly where in the evolutionary lineage one entity was a chicken or chicken egg while its progeny wasn’t; but even then, perhaps you can say that, because if you did have a definition that precise then the chicken egg would necessarily come first, it logically follows that the chicken egg must have come first, or at least that the egg is the one that came first insofar as there is any logical answer to the question.
- One thing people don’t seem to think of when considering this classic question is that, at least the way it’s worded, it doesn’t actually specify that the egg in question is a chicken egg; it gives a choice between “the chicken” and “the egg.” Eggs are much more general concept than chicken eggs specifically (e.g., even lizards have eggs), and whatever species the chicken evolved from obviously must have laid eggs, therefore the egg came first.
- You eat eggs for breakfast and chicken for dinner, therefore eggs come first.
So as you can see, the egg comes first in all possible ways of looking at it. =P