There are currently about 41 million Americans without Health Insurance. Uninsured people are less likely to seek treatment for medical issues, are more likely to die as a result of treatable health problems, and are more likely to use the ER instead of a physician's office when a problem does arise (often because the patient did not seek treatment when the problem first arose, and it subsequently got worse).
Almost every state in the US requires drivers to carry automobile liability insurance. Yet there are no government mandates requiring citizens to have basic, catastrophic health insurance coverage. In the case of liability auto insurance, it makes sense to require that all drivers be insured. That way, if you run into someone else, and you're at fault, your insurance company will compensate the victim for their loss. However, it is not mandatory to have comprehensive/collision insurance on your own vehicle (unless you have a car loan, then the lender can require it, but it's not a government regulation). This is because you can choose what you want to do with your own vehicle, and if you would rather take the risk of losing it in an accident, you can. Some people might say this is the same as health insurance, ie. it's your health, so if you don't want to insure it, you don't have to. I disagree. The problem with this logic rests in the fact that our society does take care of people in a life or limb health emergency, and payment issues come later (as it should be). If you are in a bad accident and transported to the ER, you will be admitted and treated to stablize your condition before payment issues arise. This is the only humane way to handle health crisis situations, but it leads to a problem if the patient is uninsured. If the patient is unable to pay his or her medical bills, the costs are passed on from the hospital to paying patients and insurance companies. This is part of the reason that health care and insurance costs are so high - someone still has to pay for the services provided to people without insurance or a means to pay their medical bills.
I believe that basic medical insurance should be mandatory, and proof should be required, perhaps in order to obtain or renew a driver's license. This could be a government program, or could continue to be through private insurers. Medicaid is available to the lowest-income sector of the population, and the rest of us could choose high-deductible, no-frills coverage to get us through an emergency. If everyone were paying into the insurance system, perhaps each person's premiums would be lower, since the cost of providing medical services to Americans would be spread more evenly over the whole population, rather than falling only to the insured members.