Bob, in my state (and probably all states) everyone who owns a car is required to have auto insurance. But my employer doesn't provide this insurance- I buy it myself- because I know I have to have it. I don't see why it should be any different with health insurance. The main thing is to get that insurance down to rates that everyone can afford.
There are two big problems with our current system- first- private insurance companies must make a profit- and that profit (and its a big profit) comes off the top of the premiums that we pay before they do anything for our health. Another thing that makes our healthcare by far the most expensive in the world is the fact that doctors and hospitals have to deal with a plethora of different insurances companies- each with different forms, rules and procedures- some doctor offices have more people handling insurance paperwork than dealing with patients.
By having a government run uniform non-profit single payer system solves both of these problems.
Of course- you are going to hear of horror stories about bad patient care in other countries- well there are plenty of horror stories that happen right here under our system! The fact is- most people in Canada and European countries who have a government run system are happy with their service- often live longer than Americans- and don't pay nearly as much for medical care.