There are some conservatives who support single payer, or at least a variation of single payer whereby the government funds catastrophic coverage for all and/or picks up coverage for those with pre-existing conditions.
Remember that this is all about health care financing, not health care per se. Who pays? And how can the cost of health care be made more affordable? And how does the means of financing affect the quality and quantity of care? Remember that while the American model leaves millions uninsured, most of us who are insured benefit from coverage of innovative medical treatments and drugs that wouldn't even exist were it not for the incentives that exist here for development of such technologies.
A hybrid approach, where catastrophic coverage is provided for all through general taxation and private insurance is left to flourish to fill in the gaps, has merit IMO. So would general taxation addressing coverage for "uninsurables" with pre-existing conditions. With these costs taken out of the claims pool, insurance covering everything/everybody else suddenly becomes more affordable, without sacrificing the incentives that allow medical innovation to flourish. Are there drawbacks? Sure - this stuff is complex, and the devil's in the details. And single payer is expensive, requiring either higher taxes or the elimination of deductions for the cost of private or employer-provided medical insurance.
I won't go into any more detail; not sure if anyone cares anyway if an ideological label can't be slapped on it. But for us wonks, this stuff is fascinating, and the bottom line is that there are myriad approaches to financing health care that have different impacts on the various actors - consumers, providers, insurers, employers - with a stake in the outcome.