It is an entity recognized by the state. It is not an institution of the state.
Although it has many public requirements, it is a thoroughly private entity. In fact, in the eyes of the law, a corporation is viewed just as you and I are - as an individual entity.
CA....
Many of us are already not considered private individuals, and when National Health Care becomes reality it will be a short step to declaring all citizens property of the state. All institutions are institutions of the state; the Gov't itself is an institution of the state. Individuality is a quality not related to public or private.