Because the market might encourage competition and COULD force down costs that way... you also have competing pressures to keep costs down by not covering people who would cost too much to cover. Look at airlines? more people want to fly, but in order to cover costs the quality of service drops and the inclusions become fewer.
Plus, healthcare is an essential service, if people need it and are able to pay, they will be the ones who will benefit. What real desire will there be to reduce costs on a service that people need as oppose to want.
It's kind of amusing how quickly they'll accept lies about the standard of care we in universal system get, but are unable to accept the shortcomings in their own standards.
Ghost_white made the point about airlines.
As dead0man said, I'm not totally opposed to the idea of universal healthcare. I believe it's noble and perhaps the government should only insure the poorest citizens of the country, while others can opt in to the system. Just so long as there's an "opt out" option.