Switzerland uses a system similar to the original plan for Obamacare, where individuals are mandated to buy insurance, so if Switzerland is what you mean by "universal healthcare" then it's not only not impossible, but the USA already has a more voluntary take on the exact same concept.
The US system is more expensive partly because Americans throw a fit any time insurers try to constrain costs. The blowup over anesthetists is one example. Swiss insurers seem more able to tell doctors they're spending too much and to cut it out, without patients engaging in lobbying efforts to stop it. It's a cultural difference rather than an important difference in system design.
