I personally would like to see a government run system in the United States for several reasons:
1) It is the government's job to provide health care. In article I, section 8 of the constitution, it states that congress has the power to provide for the "general welfare" (welfare as defined as "health, happiness, or prosperity; well-being" when the constitution was written). The fact of the matter is that the federal government already does do this, but to a lesser extent via Medicare, SCHIP, and the VA. Of those, the VA affects me the most personally because they actually payed for a private flight all the way accross the country to his home for my grandfather (84) after getting a bad stroke.
2) A government run system would not be for-profit. One of the main problems with the current system is that workers at health insurance companies get a larger paycheck based upon how many people they deny coverage. Contrast that with NHS in Britian which gives doctors pay raises based upon how many people they get to quit smoking, lose weight, etc. A not-for-profit system would also streamline the administrative costs. I think that in the current system, they're around 16%, whereas in other countries they're around 6%.
3) Health care for everyone means higher quality health for you. Take any disease as an example. If everyone was doing regular checkups, which people tend not to do now due to having a copay) people would be receiving feedback more often on how healthy they are, getting suggestions from the doctor, etc. If said person became healthier, they would be less likely to contract a disease. So if he was sitting next to you on the bus, you in turn would be less likely to contract the disease, clearly making you healthier.
Personally, I would like to see a creation of a government beaurocracy that would be like an HMO, but anyone could apply and be accepted. In this system, a customer would only pay his or her monthly bill, and everything health-related would subsidized. In turn, people would go to the doctor more often, guiding them closer to a healthier lifestyle (of course limits would be placed on such regular check ups)
As this kind of system grows based on a percentage of the American population, it would begin to raise taxes slightly while reducing the premiums for the customers. This would eventually result in a gradual phase-in of such a system where the system would start off with competition right off the bat, and as it became better, more people would switch over, making it more firmly rooted.
In addition, a payment constraint could be put on its growth -- say once 20% of the population uses it, it starts taking 25% of the total payments from taxes and then a bill goes out to congress as whether to leave it as is, or continue it's augmentation of funding via taxes.
I feel that in the end taxpayers should be the primary supporters for the system (once most people start using it) because taxes and government spending are a "From each according to his abilities, to each according to his needs" method wheras insurance premiums aren't.