There are three ways to answer this:

1. There is a continuum of forms of eggs and animals in that strain's evolutionary history, that we categorize as being chicken or non-chicken. If you somehow draw a distinct conceptual line between what's a chicken and not a chicken, then the first chicken egg or chicken that meets your criteria for being a chicken egg or chicken must be the egg, because there's no genetic change in the interval from the egg being and egg to it being a chicken, but there is genetic change in the interval between the chicken and the egg it lays, because it's in the process of making the zygote that the genome changes.

Or you could say that it's impossible, or ad hoc at best, to make such a precise definition of chicken or chicken egg that you can delineate precisely where in the chain of evolution one thing wasn't a chicken or chicken egg and its successor wasn't.. but even then, perhaps you can say that, since if you did have a definition that precise the chicken egg would necessarily come first, it logically follows that the chicken egg must have come first, or at least that it is the one that came first insofar as there is anly logical answer to the question.

I guess if you don't draw the line in terms of genome, though, then an animal could fit into your definition of chicken before an egg fits into your definition of chicken egg. So it kinda all depends on your definitions.

2. One thing people don't seem to think of when considering this perennial question is that it doesn't actually specify that the egg is a chicken egg. It just says, "the chicken, or the egg." The egg is much more general than the chicken, e.g. even lizards have eggs. So whatever species the chicken evolved from clearly laid eggs, therefore the egg came first.

3. You eat eggs for breakfast and chicken for dinner, therefore the egg comes first.

As you can see, the egg comes first in all possible ways of looking at it. =P