There will be rationing under a government health insurance system. That is just simple economics. If you give away a valuable thing for free, then demand will exceed available supply. The government will have to allocate health care resources according to some formula, and I guarantee you, this will be extremely controversial and morally agonizing.
Proponents of a government health plan point out that in a free market rationing is also done, but it is done by price rather than by bureaucrats and lawmakers. The price rises until demand and supply are matched, and those unwilling or unable to pay that market clearing price will receive less health care.
This is certainly true, but rationing by price is more efficient and morally sound. Those that value health care a lot will be more likely to obtain it than those who don't value it very much. It is important to remember that health is not something everybody values equally. And pretty much everybody makes tradeoffs between health and money. That is why some people are willing to be coal miners or firefighters or football players, or are willing to work overtime with little vacation or even rest. It is also why some people don't eat healthy foods or exercise or stop smoking.
Some people care a lot about having (painful) surgery so they can continue to play tennis. Others not so much.
When you have rigid rules divvying up resources, you inevitably wind up giving something the government bureaucrats say is worth X to somebody who values it at X/10. Meanwhile, a person who values that thing at 10X must go without.
Perhaps most important of all, rationing by price will lead to more production of the health care services that are the highest price and therefore in the most demand versus available resources. In the long term, a market-based system will produce vastly more health care resources for everybody.
It's amazing that some people (read Barack Obama, Nancy Pelosi, Harry Reid, and other Democratic leaders) so easily forget the lessons of the Soviet Union and Communist China. Government control of the market stifles economic growth. It has been shown empirically that government bureaucrats simply can't do as good a job as the free market at pricing goods and services, providing incentives, and stimulating increases in productivity.
Some people want to believe health care is or should be exempt from market forces, but it isn’t, and it shouldn’t be.