In a completely free market, you'd have all kinds of insurance products covering different illnesses. Just like with most other products, a private system would be much more efficient than a state run or state regulated one.
If the issue is that some people would be unable to get a health insurance, or would only be able to get a limited one, than there is a much better way to solve this problem. How about just giving those the people the money to buy an insurance? The cost to the taxpayers is the same, but without the distortionary effects of a state regulated system.
No, it wouldn't. The US is a prime example that it simply is not. Healthcare is more expensive than it is in western Europe and it's worse too. It is in fact the prime example of what can go wrong with a free market and you'd have to blind not to see it.
Health insurance as such would be impossible for a lot of people in a free market. I'll try to explain this you in a few logical steps:
1. An insurance policy is an aleatory contract, this is a contract in which the performance of one or both parties is contingent upon the occurrence of a particular event. This means there is always a risk involved. For the guy paying for the insurance, it's the possiblity that he may never need the insurance and thus lost money paying the premiums. For the insurance company, it's the possibility that the event actually takes place and they have to pay;
2. Everyone needs healthcare at some point in their lives. In an ideal world you go for routine checkups with doctors and dentists at least once a year. This means you can't really get an insurance policy for this type of thing, because the so called uncertain event
will always take place.
3. That's why in a lot of European countries you have a socialist healthcare system. This means that the system pays for routine check ups for example. If you are hospitalized though, you get a basic care package, but on top of that you can get an insurance policy for any additional losses you might face. For example the system will pay you 1,000 Euros a month because you can't work, but you can get an insurance policy that pays you 500 on top of that, because you are used to making at least 1,500 Euros. The reason you can get an insurance policy for this type of health care is that the event is not guaranteed to take place. Everyone needs routine check ups, not everyone gets hospitalized on a yearly basis.
4. However, there comes a point when even getting hospitalized yearly becomes inevitable. If you've had cancer before, if you're overweight or even if you're just old, the risk of you getting hospitalized is very high. It is in fact so high that an insurance company won't offer you a policy, because the only reason they offer you one is that they gamble on the fact you won't get sick. If they are pretty sure you will get sick, it makes no sense to offer you a policy. At this point, you have become uninsurable. Given enough time, this will happen to everyone.
The consequence of all this? Well, easy. In a completely free market insurance system, people die from illness or old age out on the streets. That's the logical consequence of what you are proposing.
Which is exactly why not a single developed country
in the history of the world has considered your system viable.