One great thing about single payer health care is that it obviates the need for most auto insurance. After all, the three main components of auto insurance are liability, which pays for medical costs, lost wages, and disability of the other driver if you cause an accident; un/underinsured motorist, which pays for medical costs, lost wages, and disability for you if the other driver causes an accident and has no or too little coverage; and personal injury protection/medical payments, which are typically non-fault, and so can take care of immediate care.
If all medical bills are paid by the government, those three become unnecessary. Only proper disability coverage is left out (SSDI is really not sufficient).
What's left? Just property damage, and there's a perfect way to deal with this as well, something that's been introduced in a couple states, but has yet to be implemented. It's non-fault, pay at the pump insurance. If it were just property damage, putting a nickel tax on every gallon of gas and using that to create a fund that all property damages would be paid out of would finally eliminate the need for any auto insurance. And hey, since single payer is a long, long way off (sadly), we could do as previous initiatives have done and make this a $.50/gallon surcharge and just cover liability as well.
It'll be cheaper than what you currently pay, again for many of the same reasons single payer health care is cheaper than our current system. There's no profit motive. The need for investigations to determine fault disappear. Everyone's paid out of the same fund, and the medical bills are right there in front of you, so the haggling over damages disappears, again thanks to the lack of a profit motive.
And no one gets screwed by auto insurance companies. I just quit the industry, as an agent for almost three years, and my experience has left me convinced that the auto insurance industry cannot be relied upon to treat clients equitably. My father is a personal injury attorney, and therefore, his entire practice rests on proving to judges and juries that insurance companies have acted in bad faith and are reimbursing for damages far less than they should. He's very successful.
Also, people who cause accidents wouldn't be financially ruined. Like I said, nearly three years as an agent, and perhaps 10% of those I met with had anything near proper limits of liability. Typically, they had state minimum, which would pay $25,000 per person, when the average moderate injury accident in my county was $147,000-$257,000. The insurance company would send a check to the court for $25,000, and the rest was on the poor guy whom no one had ever explained insurance to. They lose 25% of their salary until retirement, and often their homes and retirement.
Finally, though it would save money, the sticker shock on a gallon of gas would probably still have the virtue of encouraging people to drive less.
Too bad that we have two powerful lobbies, the insurance industry and the trial lawyers to go against. If passed, this would basically put my dad out of business, and would remove the second most profitable line of insurance from private sale. Still, dad's only a couple years from retirement, so I guess he'd approve.