The question of whether the chicken or the egg came first is incredibly old. Hints of the underlying problem appear in works by Plato and Aristotle, and the issue itself is directly addressed by Plutarch.
First, the problem needs to be set up properly, to prevent flippant answers that don't really tackle the question.
"Dinosaurs laid eggs millions of years before chickens existed, so the egg came first"
Not much of an answer, really. Implicit in the question is that we're talking about chicken eggs here. So let's set it up as "Which came first, the chicken or the chicken-egg", if that makes everyone happy
"The Bible says that God created animals (see Gen 1:21), so the chicken came first."
In addressing the key issues, this isn't a bad response actually. The deeper question - how do we resolve causal loops to origins - is rather neatly solved by saying "We don't. God does, by starting things off."
However, it requires a belief in God as the Prime Mover. As many people aren't happy with this, let's skip this answer and press on with the question.
"Evolutionary theory tells us that at some point, a non-chicken creature laid an egg which grew into a chicken. This means that the chicken-egg came first."
Hmm. This answer begs the question (which doesn't mean what you think, check the writeup). What the person who responds this way has done is assume the answer to motivate the answer, by defining a chicken-egg as "The thing a chicken hatches from". But why not define it as "The thing a chicken lays"? Then the answer to the question, using the evolutionary logic, is that the chicken came first.
But we haven't really got anywhere. Either we can get stuck in a Sorites Paradox, arguing about when a chicken becomes a chicken and an egg becomes a chicken-egg, or we pick one of the two arbitrarily, which is much the same as answering the question at random. This avenue seems fruitless
"But if the chicken came first then where did the egg come from. And if the egg came first... You've confused me, and you aren't letting me give a good answer!"
Aha! Now you're getting it. Whether the chicken or the egg came first doesn't bear answering, it's not that sort of question. Rather, it's used as symbolic of the problem of resolving causal loops. The Creationist and Evolution responses refuse to accept the reality of the case by finding get-outs. That may be the way to go, and that there are no examples of the chicken-and-egg problem in real life. However, they seem to breach no logical rule. A modern example occurs in certain forms of time-travel scenario.
So next time someone gives a throwaway answer to this old puzzler, don't let them get away with it.