Coming to this question a bit late, I would like to provide a slightly different perspective.
Private vs. public sectors and national accounts
Part of the problem is purely an accounting issue: If healthcare (or anything really) is provided by the private sector, it's “easy” to record its price. If you tally all insurance premiums, you can say that the health insurance industry produces X billion dollars worth of insurance. Similarly, if physicians and other healthcare professionals work independently, each consultation is also a market transaction.
But if healthcare is free at the point of use and funded directly by the state (e.g. the NHS in the UK), what is it worth? Intuitively, you can easily see that it's worth something. One way to deal with this when computing the GDP is to consider the costs involved but aren't lower costs a good thing? If patients get the exact same service, it does not make sense to consider the country as a whole poorer! And, as @Philipp explained, even without private insurance or even private-sector healthcare at all, there would still be many people working in this area one way or the other.
The same problem shows up in completely different contexts. For example, are Europeans poorer because they tend (on average) to have more free time than North Americans and to cook for themselves instead of eating out? There are many things to be said for or against each approach (and also on the reasons behind the difference) but the answer is definitely not obvious.
Protecting existing businesses and innovation
Beyond these accounting difficulties, it should however be obvious that protecting the bottom line of each and every company (or even of entire industries) is not beneficial to ‘the economy’. If you look past healthcare and the ideologically loaded public sector/private sector alternative, you can easily find other examples of this phenomenon.
For example, consider horse-drawn carriages or wood for heating. Not so long ago, entire industries existed to provide these but they were displaced by other operators who offered better alternatives. Many of the individuals involved certainly suffered from the change and the transition wasn't necessarily easy but the rest of the economy profited through better transportation and cheaper energy sources.
To the extent that the public gets the same service, getting rid of a middle-man is a non-issue. Free-market advocates would probably argue that the public is unlikely to get the same care in a single-payer system but they should be even less concerned about the effects of the reform on the rest of the economy because adapting to changes like that is precisely what markets are supposed to do better than centralised organisations.
Or to put it more provocatively: If you think it's not the government job to pick winners and losers and support companies like Solyndra, why should you be worried about the financial health of private insurance companies? Once you have decided on the general framework, you can let the market work its magic.
Health insurance and overhead
So the real question is whether the new system (or any other alternative) is able to deliver the same level of care or at least broadly similar health outcomes with similar costs. And the data does not look too good for private insurance. Even better managed private-insurance-based systems like those of the Netherlands or Switzerland (which are doing better than the US in many respects) tend to cost a bit more than systems based on public insurance and mixed delivery (a common model in Europe) or mostly public healthcare (e.g. the UK).
The Dutch or Swiss systems are not particularly bad either so you can still make a case for private insurance on practical or philosophical grounds but they don't have any obvious superiority in terms of costs or outcomes (it's more difficult to make a case for the US system, though, it's just that much worse than anything else you can find in high income countries).
Importantly, insurers manage the flow of money and can thus influence other parts of the system but they don't deliver health care. And if they keep more money for themselves (which can easily become mere rent seeking), the public (patients/insurance buyers or their employers, ultimately the rest of the economy) or possibly health care providers have to make up for the difference and would have less money available for other things.
Even in countries with single-payer systems, most of the money simply goes to the healthcare providers themselves, which are often private businesses or independent practitioners. The organisations taking care of payments might conceivably be so poorly managed as to impair the efficiency of the system as a whole but in terms of raw overhead, they are not obviously less efficient than private insurers, in practice.
Since, generally speaking, costs and health outcomes are not dramatically worse in public insurance systems, the extra overhead created by private insurance (including e.g. advertising) is arguably a net loss for the rest of the economy.
The dirty little secret behind all this is that insurance companies are really private bureaucracies. It's easy to talk about competition and markets on an abstract level but in practice, private insurance is often just as rule-based and unpleasant to deal with as the worse state-run organisations.
It's relatively easy to design a system with some space for for-profit insurers (even France or Germany have some) but that alone does not magically create a competitive or efficient healthcare system.