Hi Tammy-
For your question about a list of NY state health insurance companies, I don't have one, but I suggest Googling to find one.
For your question about NY residents having to having to be enrolled in a NY state insurance plan...not sure what you mean here...for instance, there are people in NY state that work for companies that are headquartered out of state, and their insurance plans are offered through an insurance company that is out of state, like Anthem BCBS. So I need clarification on that point.