Interesting question. Please note that inside most chickens is an egg waiting to be laid, and inside most (fertilized) eggs is a fetal chick. So I think it's more of a "both/and" rather than an "either/or" question.
But what's with that salmon migration?
Which came first the chicken or the egg?
I've found three answers:
1. Grammatically speaking the chicken came first, as I typed to word chicken before the word egg.
2. Evolutionarily speaking the egg came first, since birds evolved from egg laying reptiles (so the egg existed before the chicken).
3. Cosmologically speaking neither the chicken or the egg came first, because both are make of matter; And all matter was created at the same time with the big bang.
Can anyone think of any other possible answers?
I say answer 2 is correct. A reptile could have an egg with a large genetic defect that became the first chicken. That's what evolution is all about.
Actually, if were going by evolution the chicken came first
If we are talking about the usual meaning of this question, and not just chickens and eggs, we have to count the reptile or whatnot that laid the egg as a chicken.
Scientists think that life got started by all the right ingredients coming together and then getting hit by lightning. This life, even if not created in the usual way, is the chicken.
So the chicken came first
P.S. Sorry for the run on answer, you hit one of my specalists topics (that is what they are called, right?) and I couldn't resist joining in.

From Wiki
Assuming a chicken egg
In this case, the egg is assumed to be a chicken's egg. This is an obvious assumption since the question itself implies a link between the two.
If one assumes the egg to be a chicken egg then one must define what a chicken egg is:
If: A chicken egg will hatch a chicken
Then a bypass is allowed: An animal that was not a chicken laid the chicken egg which contained the first chicken. In this case the egg came first.
If: A chicken egg is the egg that a chicken lays
Then a bypass is allowed: A chicken (that hatched from a non-chicken egg) laid an egg (a chicken egg).
If: A chicken egg will hatch a chicken and was laid by a chicken
Then there may be an error of definition. If the definition of "chicken" used does not refer to "chicken eggs," then the chicken must come first, because without chickens there cannot be any chicken eggs.
This is like one of those nonsensical philosophy courses: DNA proves where the egg came from as well as the chicken
My I summarize the outcome of my personal research:
1. First of all the rooster came.
2. The hen doesn't necessarily has to have an orgasm to get a fertilized egg.
3. You need a fertilized egg to get a chicken
4. In our local grocery there's first chicken and turkey, then beef, after that all kinds of salads, and in the end the eggs.
Here's another one sort of related to that: Would the Universe exist if nothing was alive to observe it?
Something can exist without it being observed. It's like asking, "When I'm alone and not looking in a mirror, do I really exist?" Answering anything other than yes to your question is nonsensical, since it implies that the universe came into existence the moment life arose, and life cannot arise without a universe for it to arise in.
Actually, if were going by evolution the chicken came first
Why?
Scientists think that life got started by all the right ingredients coming together and then getting hit by lightning.
No. Abiogenesis is still an open field.
This life, even if not created in the usual way, is the chicken.
Wrong. That life was definitely not a chicken, the same way you're not a chicken. Or are you?
So the chicken came first
No.
Here's another one sort of related to that: Would the Universe exist if nothing was alive to observe it?
Something can exist without it being observed. It's like asking, "When I'm alone and not looking in a mirror, do I really exist?" Answering anything other than yes to your question is nonsensical, since it implies that the universe came into existence the moment life arose, and life cannot arise without a universe for it to arise in.
The whole universe is alive but I've elaborated that in the thread "origin of life" so I suggest not try to agree or disagree on that in this thread now.
The whole universe does not comform to the definition of life.
On Natalie's question no "yes/no" answer can be given. If you want an answer, no is closer to truth than yes.
That implies that before there were observers, there was no universe. But before there was a universe, no observers arise. So if there's no universe before there are observers, and no observers before there is a universe, there would be no universe and no observers at all.
You're talking nonsense.
Reality (the whole) is one. If we describe it in 3 elements: the observer (Simen), the observed (the Universe) and the observing (the process).
I am part of the universe. The boundary between me and the universe is purely artifical, because the same particles make up both me and the universe, and there's no telling which particle is me and which is universe.
Taking 1 of these 3 elements away (for example: no observer) is not possible, because reality is 1 whole.
Wrong. When I die, I am no more.
So the question, Would the Universe exist if nothing was alive to observe it, is absurd. I call a question absurd if no answer is possible.
The answer is yes, and you're not making sense.
Do you agree every human being is connected with the rest of the world? That existence is a whole?
A necessary consequence of the fact that the boundary between me and the universe is artificial is that the artificial construct "me" is connected to the rest of the world, since I'm really part of the rest of the world.
Is this question that I'm about to post here really realistic? Does time/space have an end?
Is really realistic something like true truth? 
... if your definition of a chicken egg is that it have chicken DNA. But if you'd call it a reptile's or some other bird's egg because they born the egg, the chicken would have come before the chicken egg.
True, it seems like a simple enough problem. The DNA came before either the egg or the chicken since the natural order of things is to evolve from simple to complex.
But, that is assuming a Darwinian model where evolution happens over time, or in fits and jumps over many eons.
It could be much simpler in that somewhere out there a chicken just simply laid and egg a few moments ago.
Therefore, a teleological argument might ensue where God on high created the chicken and chicken begat son of chicken.
B"H
"Egg" has three letters. "Chicken" has seven letters. Both 3 and 7 are primes. However, 3 comes before 7. Thus, if we count the letters in the English language, "egg" does come before "chicken".
I haven't read all of these, so this possibility might have been covered before. My answer may not be the most correct answer. However, I am willing to bet that it is the silliest!
All the best.
If you believe in Creation, the chicken came first.
If you believe in evolution, the egg came first-- the chicken evolved from something.
i'M SURE THE EGG CAME FIRST.
Coming back to this one I'm thinking the dino laid the original egg and the successive progeny got all downy like and eventually sprouted feathers and perhaps flew for a bit and then landed and crossed the road.
But the chicken was in an egg before it hatched, right?