How would free healthcare in the United States change the expenses of people in the lower class, middle class, and upper class?

The first issue here is that "free" health care isn't actually free. Hospitals, doctors, and medications all cost money. The actual issue is how those costs are paid for and by whom. In some countries, patients pay health care providers directly. In other countries, people tend to buy insurance and insurance companies then pay all or part of medical costs. The standard model in most developed countries is some form of national healthcare, usually with a single payer model, in which the government collects taxes and uses tax money to pay the health care costs of all citizens and permanent residents.

The current system in the United States is actually a mixture of government (i.e. taxpayer) funded health care available to veterans, lower-income families, and the elderly, for-profit insurance, and direct payment to providers. 

What changes people will experience in health care-related costs if a single payer system was instated would depend on implementation details. For veterans, the elderly (Medicare eligible), and the poor (Medicaid eligible), a switch to a single payer system might not not change their costs or benefits.

For workers who already have insurance plans through work, out-of-pocket medical expenses might be lower. Currently, approximately 60 percent of bankruptcies in the US are related to medical debt; this number would decrease markedly under a single payer system. People in lower- and middle-income brackets who are not Medicaid-eligible would probably see the greatest benefits. People in the middle- and upper-income brackets would see a reduction in out-of-pocket medical expenses and insurance costs.

The question remains, though, of how such a system would be financed. It would require some form of increased taxation or perhaps mandatory insurance contributions on a sliding scale. While consolidation of Medicare, Medicaid, and VA benefits under a single payer system would result in some cost savings, some people would still need to pay additional taxes to fund a single payer system. If such a tax plan closed loopholes such as the difference between higher taxes on wages and lower ones on capital gains (which benefits the wealthy), then the wealthy would pay higher taxes. An across-the-board tax increase or fixed levy might increase tax rates for some, but this might be offset by lower medical costs. 

Realistically, the answer to this question depends entirely on implementation details; without those, one cannot know actual numbers.