Research suggests that the protein essential for the formation of chicken eggs (OV-17) which is only found in chicken ovaries. Without it, the chicken egg shell could not be formed. So, without a chicken, you technically can't get a chicken egg. But this all depends on the nature and definition of a chicken egg in the first place. After all, chicken egg is an egg laid by a chicken or one that simply contains a chicken? Obviously, the OV-17 bearing chicken had to come from somewhere.
During reproduction, two organisms pass along their genetic information in the form of DNA but the replication of this DNA is never 100% accurate and often produces minor changes for the new organism. These small mutations in DNA over thousands of generations create new species 'but these genetic mutations must occur in the zygote, or initial cell.
So, a creature very similar to a chicken, which we could call a proto-chicken, would have mated with another proto-chicken and because of a small genetic mutation, created the first chicken which grew in an egg. So, the egg came first?
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Well, Team Chicken might argue that this was simply a chicken growing inside of a proto-chicken egg however; no one mutation can ever really constitute a new species. Even though we humans like to classify all creatures into different groups and names this is based on how things currently are and not how they were millions of years ago. The process of evolution is so gradual that no one proto-chicken to chicken birth could really be considered a new species at the time much like how dogs have come from wolves. As humans began to interact with and domesticate wolves. There was no one single point where a wolf gave birth to a dog but rather, as particular traits came about from selective pressures such as choosing wolves that are not afraid of humans or ones that were less aggressive over many generations we can see big genetic and behavioral trait differences.