Wrong way 'round Mike though I am glad you're contemplating the position that insurance is bad not good.
What everybody should have is free yearly checkups. Then the patient can discover what is wrong. The insurance, if it exists, should be for anything else.
People can choose to call it quits at 80 as has been suggested, or pay for all the goodies and anything in between. Obviously, one's insurance rates will be higher if you are overweight, smoke, don't take medicines etc etc... The patient might be able to find specialists that will work for free. The relationship will be a lot different if a patient is getting free care (and had to ask for it), than expects the "rich" doctor is being paid by the government and they can do whatever they want and still sue for a big payday.
People may be a lot less inclined to get every test in the book if they have to pay cash. People will become accustomed to "thinking ahead"
and having a savings account. People may actually start to take care of themselves.
The government would have to come to grips with rationing care for all those that don't have income (something they are not willing to do because of the political ramifications - "I'm sorry Ms Jones you can't get dialysis" will not go over well at the next election).