When a person who has no insurance needs care, he or she eventually goes to the emergency room. This is not a cheap or efficient way to provide care. It's bad for the patient in a lot of ways, because the waits are long, the overhead costs high, and the training of the physician is focused on traumas and other urgent care matters.
Uninsured people in my experience know perfectly well that emergency rooms are for people that are really sick. Their means don't extend to paying for non-urgent care. So they wait until things are seriously life-threatening before they make the decision to seek care. Treatment eventually winds up costing more, and often leads to an avoidable hospitalization.
Then there's the other costs. The uninsured usually can't pay their hospital bills, so we physicians wind up writing off our time in these situations. The hospital admin people are obliged to make an attempt at collection, often futile, but if not almost certainly financially devastating to the patient.
Under a single payer system, many of these problems go away. The collection problem is an obvious gain, but a less obvious one is the cost savings due to providing non urgent care where it's needed. I called Cottage Hospital local to me where I live and they tell me that an ER non-urgent care visit costs 2.4 times what an ordinary office visit does.
This isn't a scientific survey, but my town, though a small affluent one, is reasonably representative of the US in most of the statistical measures. We have about the same proportion of uninsured, anyway.
When calculating cost savings from hypothetical single payer health care systems, I have been assuming that the uninsured population is broadly similar to the Medicaid population excluding the disabled and elderly. There are about 40M children and adults covered under Medicaid, and they're consuming about 1850 dollars per capita according to the Kaiser Family Foundation. Covering our uninsured at this rate would cost us nationally about 83.4B. However, the savings in moving non-urgent care from the ER and avoiding some hospitalizations reduces this value. By assuming that non-urgent ER visits are eliminated, taking no benefit for better provision of ER care to actual urgent cases, and reducing hospitalizations of uninsured patients by 5%, a savings of 27.9B, or about 1/3, can be realized in a single payer system. Thus, the additional cost of insuring the uninsured is less than it might appear, about $55.5B, or 1233 per capita.
There's an additional benefit that can't really be quantified but matters a great deal to me as a matter of equity. In Canada, the most similar population to the US in the world, the life expectancy is two and half years longer and infant mortality is markedly lower. If you exclude the poorest twenty percent of Americans from health statistics, the two countries have the same life expectancy and infant mortality. We pay a terrible price for our uninsured, and besides being foolish, it's also wrong.