As a Catholic, you should know that there was no death before the fall. Adam and Eve were vegetarians. Meat wasn't given as food until the time of Noah. So Biblically the chicken definitely came first.

Of course, that's assuming we're talking about the first chicken ever versus the first (chicken) egg ever. The question doesn't actually say that, however, and there are other ways in which it might be reasonably interpreted. (Unless there is some additional information of which I am not aware in the context surrounding the original question. But if so, the poll did not supply that information.)

If we're talking about any random chicken one might happen to encounter, and the corresponding egg out of which it hatched, then that complicates things. Then you have to look at things like whether the chicken is a chicken before the egg is fertilized, and what constitutes an "egg" anyway (does it have to have a hard shell yet, or will a couple of cells do), and so forth.