"Did I just blow your mind?" -Sterling Archer

Using evolution, the egg was likely an adaptive mechanism to cope with pressures faced with external fertilization. Animals that couldn't produce a hardened shell likely had a time procreating and producing offspring. Therefore, I'm beating that the animal came before the egg lol

In nature, living things evolve through changes in their DNA. In an animal like a chicken, DNA from a male sperm cell and a female ovum meet and combine to form a zygote -- the first cell of a new baby chicken. This first cell divides innumerable times to form all of the cells of the complete animal. In any animal, every cell contains exactly the same DNA, and that DNA comes from the zygote.

Chickens evolved from non-chickens through small changes caused by the mixing of male and female DNA or by mutations to the DNA that produced the zygote. These changes and mutations only have an effect at the point where a new zygote is created. That is, two non-chickens mated and the DNA in their new zygote contained the mutation(s) that produced the first true chicken. That one zygote cell divided to produce the first true chicken.
Prior to that first true chicken zygote, there were only non-chickens. The zygote cell is the only place where DNA mutations could produce a new animal, and the zygote cell is housed in the chicken's egg. So, the egg must have come first.

Explained really well below

The theory of evolution states that species change over time via mutation and sexual reproduction. Since DNA (deoxyribonucleic acid) can be modified only before birth, it can be argued that a mutation must have taken place at conception or within an egg such that a creature similar to a chicken, but not a chicken, laid the first chicken eggs. These eggs then hatched into chickens that inbred to produce a living population. Hence, in this light, both the chicken and the structure of its egg evolved simultaneously from birds that, while not of the same exact species, gradually became more and more like present-day chickens over time.

However, no one mutation in one individual can be considered as constituting a new species. A speciation event involves the separation of one population from its parent population, so that interbreeding ceases; this is the process whereby domesticated animals are genetically separated from their wild forebears. The whole separated group can then be recognized as a new species.

The modern chicken was believed to have descended from another closely related species of birds, the red junglefowl, but recently discovered genetic evidence suggests that the modern domestic chicken is a hybrid descendant of both the red junglefowl and the grey junglefowl. Assuming the evidence bears out, a hybrid is a compelling scenario that the chicken egg, based on the second definition, came before the chicken.

This implies that the egg existed before the chicken, but that the chicken egg did not exist until an arbitrary threshold was crossed that differentiates a modern chicken from its ancestors. Even if such a threshold could be defined, an observer would be unlikely to identify that the threshold had been crossed until the first chicken had been hatched and hence the first chicken egg could not be identified as such.

A simple view is that at whatever point the threshold was crossed and the first chicken was hatched, it had to hatch from an egg. The type of bird that laid that egg, by definition, was on the other side of the threshold and therefore not a chicken—it may be viewed as a proto-chicken or ancestral chicken of some sort, from which a genetic variation or mutation occurred that resulted in the egg being laid containing the embryo of the first chicken. In this light the argument is settled and the egg had to have come first.