This depends on what you mean by "hole".
In the topological sense, eggbug, like most animals, has one hole (from mouth to anus) and the entire animal is built around that single hole.
In a more colloquial sense, eggbug has innumerable holes: Every part of the surface of the main body of the eggbug is covered in tiny pores, within which resides more eggsbug, each of whom are covered in the same sort of pores, and so on and so forth until infinity in both directions.
Some people speculate that our world exists outside of an eggbug pore, and that we are witnessing the root eggbug, but this is, frankly, nonsense.