This seems to be a big topic right now, especially in the political ring. The Salt Lake Tribune recently ran an article about an idea along these lines. The article says that the insurance would actually be purchased from the private market but at affordable prices. The idea is to create a "stock exchange" where individuals would use pretax money and employer premiums to purchase their coverage. The exchange would not be a government entity but a non profit entity. The plans would also follow the individual and not change with job changes. The backers of this idea claim that it would increase competition between insurance companies which would drive the cost down. The article goes on to discuss more details and how the plan would be funded.
This was a new idea that I thought was interesting and I am curious to learn more about it. I think that the traditional idea of universal health care in theory is a good idea, but the execution is tough because it decreases the incentive for the actual health care providers. I had a friend who lived in Canada for a few years who had some pretty negative experiences with their health care system. His roommate had his scrotum bitten by a dog (this is a true story) and they had to visit 3 hospitals and wait almost 5 hours for treatment because none of the doctors knew what they were doing.
Our health care system definitely has problems and needs to be fixed. How to fix it is the real question.