Does Operation Egg Heist have a Chaos mode?

What Chaos changes beyond the difficulty is not published. Treat it as the same mode with less margin for error, so learn the standard version of Egg Heist first.

That standard version is a PvPvE extraction mode, also listed as Diamond Egg Theft. Other squads are in there with you, and beating one lets you take their items, so the threat is never only the environment. Whoever collects the central egg transforms into the egg itself in order to carry it out. You become the objective, visibly, for everyone else on the map.

Getting out is the part that punishes you. Extraction means a vulnerable trip back to an escape boat while you are the thing everyone wants.

The safe exit, and how to earn it

A Pathfinder can extract safely, but only after one personal objective is done. You either personally submit one Aniimo Egg or personally collect a set number of Eggshell Coins. Personally is the operative word. A squadmate doing it does not cover you.

The coin requirement is not a published number, and it was significantly reduced in the February 4 update, so an older figure is likely to be wrong. Plan on banking coins steadily through a run rather than counting up to a target you cannot see. In Chaos, where mistakes cost more, having that objective already cleared before the fight over the central egg starts is the difference between a safe walk to the boat and a run you lose at the last minute.

Not settled: What Chaos mode actually alters, beyond being harder, is not spelled out.