To quote from memory (may be slightly off):
"We hold these truths to be self-evident, that all men are created equal and are endowed by their creator with certain unalienable rights--life, liberty, and the pursuit of happiness." U.S. Declaration of Independence, signed July 4, 1776.
No--Americans are not the only ones with those three rights. It says right in there "all men" meaning the entire human race.
Having a government mandated health insurance plan is a loss of liberty. If I go to school, work hard, get a good job so I can afford a good health plan, I get punished because my plan will be downgraded so all others can be uplifted. Even if I can afford to pay my own way to get extra services, that will be illegal. That is what the plan Hilary Clinton designed would have done, and that is why about 70% of Americans, when they learned of its' details, rejected it. Social Security--the government mandated retirement program, is a good example of this. If all Americans had invested--in the stock market, bond market, or even just put the money in the bank--the same amount as was confiscated from them in FICA, they (and we collectively)--think of all that available capital for investment--would be far better off.
And yes, one can be poor and uninsured, and still get health care. It happens all the time. What hospitals do is write off the bad debt and make up for it by charging more to those who do pay. Sort of a hidden health insurance plan. When I was given a medical discharge from the USMC, my wife was pregnant. Her continued natal care was not allowed in a military treatment facility, and insurance would not cover it because it was a preexisting condition. Besides, I had no money for insurance. We paid the hospital and doctor sometimes as little as $5 a week towards our bill until we could pay it off. But, we paid it as we viewed it as our obligation. They doctors/hospital were glad to do it.
Many of the 43 million uninsured are also uninsured at their own choice. They are usually young, relatively healthy, single people starting out in life and do not really need health insurance. If something catastrophic happens to them it is likely to be a car wreck where they can get their auto insurance to cover it.