Presently, each insurance company and every insurance company product have pretty much unique claims & payments procedures. This requires that all "health" "care" providers maintain the overhead associated with staff to figure out how to get paid and for what. Insurance companies also have to maintain the overhead associated with claims:payment processors.
In a Single Payer system, as I and friends of mine
http://www.pnhp.org/ envision it, all "health" "care" providers remain private, all "health" insurance providers remain private too, but the government becomes the conduit by means of which claims and payments are processed, just as the government currently does in Medicare and Medicaid. Removing the overhead of maintaining claims:payment processing will reduce costs for both "health" "care" providers and "health" insurance companies. Reduced costs will be passed on to consumers and also allow health care providers to improve services. This approach will cost "health" insurance companies some jobs, which is unfortunate, but all of us are being told to adapt professionally. Look at how layoffs in IT and the auto industry have been met with exhortations for people to re-train. Also, why should insurance companies be supported by Socialism and other jobs left to dog-eat-dog capitalism?
One of the things I like best about getting the maximum amount of cost cutting available to us under a Single Payer system is that such deep reductions in overhead will make it possible for us to consider expanding services covered to cover truly preventitive and complementary medicine and therapies.