The main problem with this question is that it does not specify what type of egg. Some dinosaurs laid eggs, so there were certainly eggs before there were chickens.
But let's be charitable. The question is meant to refer to chicken eggs. In a literal sense it is about eggs, which produce chickens, which produce eggs. So which comes first? Neither or both.
It is the same with acorns and oak trees or mothers and daughters. Each generation produces the seed of the next. The chicken produces an egg, which hatches into another chicken. The tree produces acorns which grow into more trees. We produce babies which grow into adults and have children if their own.
So where did it all start?
Going back through history, we will find millions of chickens and eggs. Due to evolution, as we go back a good many years, both will look different. One question raised here is how different either would need to look in order to still be recognizable as a chicken or an egg. However the egg as a less complex shape is likely to deviate or evolve less over time than the chicken. So going backwards in time we would get to something that no longer looked like a chicken before we got to something that no longer looked like an egg.
So the egg came first.
Now you could decide you don't believe in evolution, or say that however much its form changes over time the bird is still a chicken. In this case the question is more difficult.
Let's look at the problem in another way....
We have to imagine a chain of eggs and chickens stretching back into an infinite past. If it is infinite, it had no beginning so neither came first, as there was no start to the whole thing. But this seems unlikely. It must have started somewhere. Generally those who do not believe in evolution believe in some kind of prime mover' commonly known as God. If we take this view, any religious text will tell you that the creator made living things which then reproduced.
So the chicken came first.
So if there is a God, the chicken came first. If not, the egg did. Who would have thought this would turn out to be a theological problem!
