I debated answering this or not, because the US healthcare system is enormously complex and varies to some extent by state and your age, so take any answers you get as an oversimplification.
The full cost of health insurance per month for a family of four or more would be about $2k per month, but almost nobody pays the full cost on their own. Most people get subsidized health coverage from their employer and would pay about 10-20% of the cost. Costs would be a bit lower for a childless couple or a single person.
People on low incomes get free or almost free coverage under Medicaid, which is jointly funded by the Federal government and the states and administered by the states. People 65 and older and people with disabilities get coverage under a similar program called Medicare. People who don't qualify for either can buy coverage established by the Affordable Care Act in a government-run marketplace here. Some states run their own ACA exchanges. The cost for coverage obtained on the exchanges depends on your income. The lower your income, the more the government pays.
Military veterans, of whom there are several million, get medical care under the Veterans Affairs system, which runs their own hospitals and clinics. It's comparable to the NHS in many ways.
Hospitals cannot withhold emergency care based on your insurance status or ability to pay. Hospitals and medical practices are happy to accept people who want to pay directly for care, as it saves them a lot of admin involved in the insurance billing process. 92% of Americans have some form of insurance coverage, with employer insurance covering 66% of the population and the government providing the rest. See here for more details.
On the debt collection question, it varies based on who provided the care. Some non-profit hospitals write most of it off and get government subsidies to do so. For-profit hospitals and medical practices are more vigorous about it. Generally speaking, if you can pay, they'll pursue it, but they can't get blood from a turnip, and they don't want people to declare bankruptcy, which would erase the debt.
As I said at the beginning, this is just an overview, and there's a lot more nuance.