Well, a creationist will tell you that God created the egg first and that it hatched and grew up into becoming a chicken.
A big boomer will say that in the cosmic explosion that brought in life to the planet, the most primitive forms of life were the first to appear. Since the egg is more primitive than the chicken, the egg wins again.
If you are more metaphysical than that, still the egg takes the lead because with the egg comes the evolution of the self into a whole being. Hence, the secondary nature of the chicken :)
The answer is basic, simple, provable and well-known...
The egg came before the chicken. By about 500,000,000 years
Because this whole 'egg or chicken' paradox is an old question. It is a pre-Darwinian thought experiment. And the answer to this 'paradox' was demonstrated 150 years ago when Darwin realised that species evolve.
i.e. The problem 'appears' to be that 'You need a chicken to make an egg and an egg to make a chicken' and this presents us with a logical disconnection. But only because the question implies that species are fixed...
But Species are not fixed.
The answer is simple... You DON'T need a chicken to make an egg. Eggs are not exclusive to chickens.
On an evolutionary timeline...
Fish laid the first eggs.
Then, Amphibians evolved the first land-based eggs.
Then reptiles improved on them.
Then dinosaurs had 100s of millions of years of laying eggs.
Then, eventually, birds evolved, and they too laid eggs.
So... The Chicken is an unimportant, minor bird with no evolutionary prowess or significance. It is insignificant,
But The Egg is an extremely ancient development and absolutely central to the understanding of how marine-based life managed to conquer the land and still reproduce its water-based babies. (An egg is a recreation of a water environment, for animals who evolved from water-based life but moved onto land and gave birth there.) (if you think about this until you understand it, you will be a modern, scientific thinker.)
The Egg is a fundamental anchor point of evolution. The Egg is a key to comprehending the conquest of the land by water-based creatures. The Egg is essential.
The Chicken is not.
The 'mind-bending' Chicken and Egg problem is only confusing for people who don't understand evolution. Evolution can explain the answer as simply as 1+1=2.
FACT 1) Eggs are very very important and ancient and essential to the evolution of planet Earth.
FACT 2) Chickens taste nice and are a fat, unimportant, modern bird with no influence on evolution.
Chickens did not invent the egg. Not by a country mile. And that is obvious if you read a little into evolution.
A Major Charles Darwin Fan
I think the confusion about "who came first - egg or the chicken" exists because we have tried to find an answer to the question without clarifying what we mean by either the firs chicken or the first egg. If we consider that the chicken have evolved from some other creature. Then we can say that the first chicken was the first bird with all the essential characteristics of a chicken as we know it today.
Having defined the first chicken in this way, we can go about defining the first egg of a chicken. This can be the egg that produced the chicken, or it can be the first egg produced by the first chicken. I prefer to consider the egg that produced the first chicken as the first egg. Once we accept this situation, it becomes very clear that the egg came before the chicken.