By straight-up egg I would assume he means a beaten raw egg with no liquid added, not a cooked egg. In another example, if you order a drink "straight-up," you want it without anything else added to it.
An egg wash is a beaten egg but often has a little water, milk or cream mixed in with it.
This, as is the case with many things, depends on where you are. On the west coast, at least, of the U.S., a straight-up egg is an egg that is fried and not turned over. It's the same as a sunny-side-up egg.
Given the context of a chef instructing someone to dip the chicken in the egg, it seems likely that it's a raw egg and not a cooked one, even on the West coast