First came the reptiles, then the chicken. Eggs are hundreds of millions of years older than chicken.
Obviously parents come before offspring.
The earliest form of reproduction is just binary fission - what an amoeba does by splitting itself into two. In it's simplest (origins of life) form this is just a mechanical process of a proto-metabolism (bunch of chemicals) being split into parts that are smaller versions of the "parent", e.g. some proto-cell composed of seashore froth being whipped into smaller versions of itself.
Eventually the division process became asymmetrical with the spawned child being a simpler proto-version of the adult, capable of then independently developing into the adult form. It seems life developed in the oceans before emerging onto land, so the history is probably of multi-cellular fish-precursors reproducing by spawning simpler (conceptually egg-like) versions of themselves, eventually evolving into egg-laying fish, and then egg-laying land-based animals including the dinosaurs from which birds developed.
So, maybe the best answer to "which came first, chicken or egg?" is "fish".
And of course eggs predate chicken, eggs are seriously old technology.
Back to the tongue in cheek: there is something easily tautological that chickens come from eggs. At some point a non-chicken laid an egg, and the first chicken was born from it. This is true regardless of which mutation one decides is "the" mutation that makes for the "first" chicken.
QED it's the egg
But it looks like phylotypic animal body plans may have originated in cell aggregates before the development of the egg:
https://onlinelibrary.wiley.com/doi/epdf/10.1002/jez.b.21417
Therefore instead of Chicken of Theseus, I propose Chicken of Thesaurus.
Which itself is surely evolved from Chicken of Thesea.
And, consequently, canned tuna.
Did the proto-chicken lay a chicken egg which hatched the first chicken, or did the proto-chicken lay a proto-chicken egg which hatched the first chicken?
If a chicken egg is an egg that hatches into a chicken, the egg came first.
We know that the first case is correct because an unfertilised egg laid by a chicken is called a chicken egg despite by definition not hatching into a chicken.
Therefore the chicken came first
Or alternatively we can say that the germ line went from 49% chicken to 50% chicken to 51% chicken? Biology is really a continuous state space anyway and trying to push things into categories too hard will break them.
>"In fact, the study shows that either the principle of embryonic development existed before animals, or multicellular development mechanisms evolved separately in C. perkinsii."
It all boils down to the debate surrounding abiogenesis: the constant egg-chicken (or gamete-organism) loop does not explain how life came to be, emerging from, in principle, no life. It's paradoxical, just like trying to answer "the chicken" or "the egg". Or, as you seem to prefer, "the reptile" or "the egg".