Do players turn into the egg in Egg Heist?

Carrying the egg is not an inventory slot. Your character becomes the egg, which means the trip to extraction is done in that form and anyone watching the objective is watching you. There is no way to hold it quietly and play normally.

You are not stuck in place while transformed. Jumping and dashing work as the egg. The rest of the movement set is not settled, so plan the route on the assumption that your usual traversal options may not all come with you.

Extraction is the point of the transformation, not a bonus on top of it. The egg only counts once it is out. Items and resources you pull out of Egg Heist can be sold afterwards for eggshell coins, and there is a bulk sell-all interface for clearing a full run's haul in one go.

What you are extracting through

How dangerous that run out is depends on which version of Operation: Egg Heist you are in, and the format is not settled. It has appeared as a competitive race between teams, as PvP played in teams of three, as a real-time PvEvP mode in the Lost Isles where three-player squads fight monsters and rival players while gathering eggs and resources, as an objective-holding match between two sides, as up to six three-player teams contesting a single objective, and as a cooperative version with no PvP combat at all. In the versions with other players in them, becoming the egg makes you the target. In the cooperative version, it is just the delivery step.

There is also a Chaos mode, a higher-difficulty version of the same operation. The mode was introduced in the third beta, so the shape you meet is build-dependent.

Not settled: What you can actually do while transformed is the open part. Jumping and dashing are in. Whether the rest of your normal movement carries over is not pinned down.