You don't have to keep working. Workers compensation, life insurance, long-term care policies all exist.
There's really not much different in what's available, only who's responsible for it. In America, you earn and keep more and it's up to you to allocate properly (or not). In other countries, the government does it all for you to ensure there's a base level of support for everyone.
It's not a question of availability it's a question of price.
If you need to pay for a very expensive medical insurance then you can only afford it while employed which you won't be if you have health issues that prevent you from doing so or get fired.
That is not how it works at all. America has both public funded (federal Medicare, state-level benefits, etc) and private insurance (either personal or group policies through your employer).
There are plenty of choices for continued care and coverage, as long as you plan for it. That's the major difference. It requires you to actively plan for your retirement and old age. Other countries just do that for you in exchange for a heavy tax.
This US healthcare meme is tiring and uninformed. A small minority have financial hardships due to medical emergencies here, and they for the most part aren’t poor, who get healthcare for free.