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There are several countries in which all citizens have a right to healthcare that is paid for by the government, e.g. in Canada.

Yet in some of these countries (again noteably certain parts of Canada), there are further restrictions on the sale of private health insurance. For example, private insurance can not be used for procedures that are available via the public healthcare system.

For example, prior to the 2005 Chaoulli v Quebec (AG) case, the Quebec Health Insurance Act and the Hospital Insurance Act prohibited private medical insurance in the Canadian province of Quebec.

What is the rationale for such restrictions?

Universal healthcare systems can be harmed by competition from private healthcare, such as when the private companies pay more and make it difficult for the state system to recruit staff. It can also make it harder for national systems to buy medicine at reasonable prices, when manufacturers can rely on the private sector paying more and people desperate for it.

There are also issues of fairness, as when private treatment is available those who can afford it can bypass queues in the state system by getting specific procedures done privately, and then return to the state system when it suits them.

Another issue is the state picking up the bill for failures in the private sector, such as a recent incident in the UK where sub-standard breast implants fitted by a company that subsequently went bust had to be fixed at the taxpayer's expense.

Universal health insurance which is affordable for everyone can only work when the healthy people subsidize the people with chronic health problems.

If you have a completely unregulated private insurance sector, then the private insurers would be free to choose their customers. Given that chance they would reject anyone with pre-existing conditions, because these people would become a net cost instead of a net benefit for the insurance company. So the private insurers would grab all the "good" customers while the government provided health insurance is stuck with the "bad" ones. That means the cost for the taxpayer to provide that health insurance would increase drastically (either because the state needs to subsidize the health insurance system or because the premiums must be increased) while the private insurers make a hefty profit.

There are basically two solutions for this problem.

Either you strongly regulate the private insurers in order to prevent them from cherry-picking their customers. You might make it illegal for insurance companies to reject customers or to take different prices for the same plan from different customers. But it's hard to outlaw things like targeted advertisement or insurance plans heavily optimized for the preferences of certain target groups.

Or you make the government health insurance mandatory for all citizens. This effectively removes the market for private health insurance and only leaves a market for "luxury" health services which go beyond the regular health care.

One reason, which is not touched upon by other answers, is that private insurance coverage would increase health care costs for everyone, including those universally covered.

One of the major points of savings in a single-payer system is the reduced administrative overhead. People automatically think of the reduced overall overhead and elimination of redundancies for a single payer vs multiple insurance companies, but what has as much impact is the reduction in administrative staff and systems needed for health care providers to handle billing.

If you have a single payer, there are only one set of reimbursements, one method of coding and bundling procedure coding for insurance claims, one set of rules and prerequisites for allowable claims. For every insurer you add, you are adding the complexity of tracking a parallel system for the differences between insurers, and then the need to track which patients align with which sets of insurance rules, so the provider and the patient don't unexpectedly get stuck with a service that can't be paid for. This administrative overhead drives up the cost of all health care for that provider, and that is reflected in the billing rates the provider will charge to all patients.

So, if you have a system that is paid for by tax dollars, allowing private insurers, to a certain degree, would require all taxpayers taking on an added burden for the benefit of that private insurer, and not just the premium paying customers of that insurance company.