Libertarianism can appear appear to endorse strong individuals ignoring the needs of the sick and the weak.
This gets to the heart of most misconceptions regarding libertarians. Apart from the occasional asshole (yes, there are asshole libertarians, just like there are asshole liberals and asshole conservatives), libertarians don't endorse ignoring the needs of the sick and the weak. We simply question that the government is the proper tool for the job.
This is where communication usually breaks down, and it is, in large part, our fault for not making our views clear. What libertarians need to understand is that most people don't see a clear distinction between government and society. To them, if "we should" do something - then "government should" do something. Likewise when we say "government should not" do something, they hear "we should not" do something. As much as it may seem obvious to libertarians that government and society aren't the same thing, it's not obvious to the rest of our current culture, and if we're to communicate our message clearly, we have to make this point clear - over and over again if necessary.
Government should do everything that needs to be done that the private sector is unwilling to do. Clearing the streets of the ill and dying poor may just be one of those responsibilities. How compassionately the job is done should be up to the voters, not the insurance lobby.