Intersting article about the Swiss Insurance system....universal cover via private insurance (basic plan offered on a non-profit basis) with supplemental plans also offered. The Netherlands and Switzerland are two models with universal cover provided via private insurance, but will mandates and regulations.
A few big differences....employers often pay premiums, but employees are taxed for this, and a fairly high level of co-pay as compared to well (over?) insured Americans. Also, docs who are found by private insurers to be over-prescribing care/drugs can be forced to pay back some of the costs! And docs make less money relative to typical wage within their nation.
So, it is possible to cover everyone via private insurance (Netherlands, Switzerland), but lots of differences between those systems and ours.