Here is a Reuter's article from August of 2008 notes the following:
Americans who go without health insurance for any part of 2008 will spend $30 billion out of pocket for health care and they will get $56 billion worth of free care, according to a report released on Monday. Government programs pay for about three-quarters, or roughly $43 billion, of the bills for these uninsured peopleAnd:
"The uninsured receive a lot less care than the insured, and they pay a greater percentage of it out of pocket. Contrary to popular myth, they are not all free riders," Hadley [of George Mason University] said.The idea of the uninsured subsidizing the insured is one that is also discussed in the book Sick: The Untold Story of America's Health Care Crisis---and the People Who Pay the Price by Jonathan Cohn. This book contains a brief but informative discussion of the current state of health care in the United States as well as some background information about how we got here (such as the beginning of group medical and the system of employer provided health insurance that still exists today).
Finally, I will admit it is difficult to exactly pinpoint what is driving cost increases in health care, though I do not believe that unpaid expenses from the uninsured represent a major factor in cost increases. Here is a link to a National Coalition on Health Care factsheet which includes this:
Experts agree that our health care system is riddled with inefficiencies, excessive administrative expenses, inflated prices, poor management, and inappropriate care, waste and fraud. These problems significantly increase the cost of medical care and health insurance for employers and workers and affect the security of families.