For some reason health insurance is one of those topics which brings the idiots out of the woodwork, largely because people seem to have little understand of what the purpose of insurance is generally. In addition, it's important to realize that we increasingly don't really have a system of "health insurance" but instead a system of "health care service providers," but we should all at least understand the general model of insurance, so we can begin there...
People want to buy insurance because they're risk averse. All else equal, people would prefer to earn $50,000 every year than $100,000 50% of the time and $0 50% of the time. Over your lifetime it'll add up to the same amount, but you'd prefer to be certain of $50,000 every year than to flip that coin every year. In rough econspeak:
U(50,000)> .5U(0)+.5U(100,000)
That is, the utility obtained from 50,000 with certainty is greater than the expected utility of a gamble which gives you 0 if the coin toss comes up heads and 100 grand if it comes up tails.
So, consider something like fire insurance. Every year there's a 10% chance that your house burns down and you lose $100,000 and a 90% chance you don't. Because you don't like the uncertainty, you're willing to pay some amount to remove that uncertainty from your life. Since an insurance company can bundle up a large number of people, as long as fires are uncorrelated events (no entire blocks burning down), the law of large numbers ensures that the total losses (equal to total insurance payouts) are essentially predictable. That is, the insurance would pay out an average of $10,000 per policy per year. An actuarially fair policy would have an insurance premium precisely equal to the expected (average) loss, or $10,000 per year, and a competitive insurance market would result in a price of just about that. The net result is that your "loss" every year is precisely certain - the value of your insurance premium. If your house burns down, you're out the $10K you paid to the insurance company, but they write you a nice check for 100 grand.
But, all that assumes the odds of houses burning down are identical and (more importantly) known by all parties. Reasonable enough for fire insurance, roughly, but not reasonable for car insurance, because some drivers are more likely to have accidents than others due to driving environment/ability/habits. If there were no legal requirements for automobile insurance, and insurance companies had no way to tell which drivers were "good" and which drivers were "bad," then there'd be a serious problem with the market. Short version is that actuarially fair premiums equal to the expected loss of the whole population of drivers would likely be higher than "good" drivers, who don't get into many accidents, would be willing to pay. So, good drivers don't buy car insurance, not because they don't "want to" but because they don't want to at a price which effectively subsidizes the rates of bad drivers. There's a market failure in the insurance market for good drivers. Bad drivers do get insurance, at an actuarially fair but expensive rate.
In practice, a combination of legislation requiring people to be some minimum amount of car insurance (for this and other reasons) and the ability of companies to charge different rates based on driving records and personal characteristics from which a risk profile can be inferred prevents this kind of market failure from happening.
With health insurance, however, the issues are somewhat different, but this post is getting too long...
