People can easily be bankrupted in the USA due to an illness. Surprisingly to some, people in countries with socialized medicine can also easily be bankrupted due to an illness. Being bankrupt in not necessarily poverty.
Almost no one in the USA lives in poverty today (mostly only the mentally ill homeless still live in poverty and they exist all over the world and are a very though case.) So if the illness is not mental illness, it will not generally drive a person into poverty in any developed country, not even the USA.
If you can go back to working after the illness you can recover from bankruptcy, most will. In the USA if you cannot go back to working you will be able to get on medicaid and SS disability. If medicaid and SS disability did not exist things would different enough that it is hard to imagine how people would get by but I would bet that they would.
The trick with all this is to help the needy while maintaining good incentives to produce. Could we be more charitable, yes but we could also be much more efficient in our charity.