I think, whether you view it from an evolutionary standpoint or a creation standpoint, we must agree the chicken had to be here first to lay the egg. From evolution, something like a chicken evolved to the point where it layed eggs and had a higher survival rate than its predecessors. From creation, God created all the animals, chickens included. The chickens laid eggs from that point on.
From the evolutionary standpoint, post #3 has it right. Keep in mind that a egg -actually an embryo- has TWO parents. In order for a new species (i.e., a chicken) to arise, the genetics of two not-quite-chicken parents has to come together just right. So the egg came first.
Just like tree came first or seed, its not so easy to answer which came first. When we look back around 4.5 billion years when life started in earth, some say life started with algae and some says life started with cyanobacteria and some says that the gases and other compounds in the environment they reacted in presence of pressure and lightning it gave rise to a complex molecule and slowly the complex molecule gave rise to an organism. With time eukaryotes slowly gave rise to multiple cell organism. Then with genetic variations which gave rise to many species. Upon reproduction they produced egg which agian gave to a new individual. We cannot predict the exact answer of evolution but we can predict that a plant came first then seed or hen came first then egg.
I agree with Post #3. The first chicken would have hatched from an egg laid by something that was not quite a chicken.
Of course, if you are religious, you can say that whatever God decided to make first is what came first. There would be no logic in this view, it would just depend on whether God decided to make a full grown chicken or an egg with a chick inside.
The Darwinian answer is different: a creature that was almost a chicken (hen) laid an egg that had a few genetic or chromosonal glitches in it that resulted in a hen being born. Think of finches: a finch with a beak that could eat a certain kind of seed gave birth (via egg) to a finch just exactly like it, except for a slight genetic change that gave it a sightly different beak, allowing it to eat a different seed. The named creature ("almost a" chicken) makes an egg that turns out to be something other than the named creature (turns out to be a chicken). By this logic, the egg had to come first. When the egg grows up, it's a hen. A non-thing can lay an egg that turns out to be a thing.