Lot's of insurance companies have non-profit branches attached to them, though, if IKEA can be non-profit i have no doubt that a health insurance company could be.
Actually, I was under the impression that alot of health insurance companies are associated with a non-profit organization at some point.
Here in Utah, Intermountain Health Care is and was a healthcare administration and an insurer. The Insurance branch made obscene profits and the health care administration side of things pumped most of that into developing infrastructure.

IHC lost their Non-profit status a little while ago, but the effect of them maintaining their non-profit status is still readily apparent. They still control more then 85% of the hospital beds in Utah. Most of the capital was being put into expanding infrastructure.

I'm vehemently opposed to private health insurance, at least as it stands right now.
i'd be more into it if it was highly regulated and monitored, but that's never going to happen. A single payer system would have actually eliminated alot of the problems with the private sector since it would have provided consumers with a default option if the private sector got too greedy/careless/etc. I tend to think this is why the public option was eliminated; it had nothing to do with public support. It had everything to do with private health insurers, who more or less penned obama-care, not wanting their industry regulated.

There are in fact many countries, such as Germany, whose health care systems rely on private non-profit insurance companies. You are required to buy health insurance. All the health insurers are non-profit. They cannot turn you down. If you lack money, you will be subsidized. It's the Obama deal made politically and economically palatable. Too bad the Americans took all the good stuff out.