Answering this question may seem like a question of biology, however philosophy incorporates everything into it's picture including this.
Now this question is only a conundrum if you assume evolution is the driving process behind the creation of animals (and dare I say it, the driving process behind everything?). Of course this is because at a glance, logically one can't exist without the other.
In reality this question opens up a bigger question .
"When does one species become another?"
This is fundamentally what we're thinking about when we tackle the chicken and the egg.
Evolution of another species requires that members of the species are separated for enough time that the mutations the different groups will experience aren't shared between each other, and each mutates in an aesthetic or cultural way that stops them being able to mate and produce offspring that are fertile.
Taking this into account, I will say the egg came first as there would have been a very delicate point where the egg of the chicken and egg of another species would have been made, where their parents in some manner would struggle to mate with each other, however the two eggs' offspring would be fully incapable of mating. Once they spread they're DNA the species of a chicken would have been solidified.
This does bring up even deeper questions which I won't mention but will be happy to discuss if anyone would like to.
What do you think? Was it the egg that came first or was it really the chicken?