As a contractor you probably should not be eligible to participate in their employer health care program either. For you to have signed up for that, somebody had to state you are an employee. Since they've already represented you as an employee, if you leave I'd feel sure they wouldn't come clean with their insurance company and say they lied the last three years just to keep you from getting COBRA. That's the only way you would become ineligible.
Won't go into the written contract issue since some other folks covered that on this thread already. But if you continue doing contract work in the future make sure you HAVE a contract that covers what-ifs and allows you a life!
Good luck!! :-)