If the companies do not earn profits, they go out of business and millions of stock owners as well as policy beneficiaries are left holding the bag.
Every major industry in America earns "billions in profits."
Single payer would give trillions to hospitals, doctors, medical facilities, ad infinitum. They all earn profits and when combined is probably trillions.
See my other post on the Workers Comp system and how medical is paid in such a 'closed' system. In such a system the profits to those providing medical care are constrained by the system in order to keep premiums 'reasonable'.
Note that in the Swiss system insurers can NOT earn a profit. An insurer does NOT need to 'earn a profit' to pay claims (they just need to take in more then they pay out with overhead for admin expenses).
This has to occur since users (patients \ claimants) don't care how much a medical provider charges since they have NO vested interest. The system sets the premiums and what medical providers earn for providing services. Note that when CA set this up it was challenged by the AMA but CA prevailed in the U.S. Supreme Court.
You should review the US Work Comp system and how it is regulated. Oh, you might wonder 'why would any FOR profit insurer wish to be in the WC system?'; The main reason is because states require them to be in WC otherwise the insurer can't sell the other highly profitable lines of insurance like auto, home, and fire.
PS: Some states also have a single-payer WC Insurance Fund that is a state agency. These monopolistic states decided to do this because they believe it would serve both the employers and the employees better than having private, for-profit, insurers.