The way I see it, we have a right to legal counsel because the absence of legal counsel could result in the government taking our life or liberty if we are denied a proper defense, whether we are innocent or not. We are specifically protected from the loss of life, liberty, or happiness at the hands of our government without due process of law.
Healthcare on the other hand has more to do with the free market. Without healthcare, we may die or get sick, but the government would not be causing our misfortunes. Nor is the government authorized to impoverish me to pay someone's medical bills. I am a charitable person, up to the point where it is charity against my will. It sounds callous on the surface, but what many don't understand is that when operating in a free market, healthcare is not expensive like it is today. The fact that everyone is basically required to have insurance anymore drives costs up. Government intervention drives costs up when we cannot go across state lines to get cheaper or more efficient healthcare. Healthcare is not a cheap commodity on it's own by any means, but it is multitudes more expensive here because of these artificial barriers we put in place. With the absence of these barriers, charities, churches, and charitable people's money would go a lot further in taking care of the few people who would not be able to take care of themselves in a free market situation.
A lot of people are quick to shoot down Ron Paul's stance on how charities and churches could help people, but that's because they dwell on how expensive things are in our big-government society, without understanding how much cheaper things would be without the big government (partly because there would be more competition, but also because of not being overtaxed). It's not hard to see their viewpoint...this isn't something many people have seen in their lifetimes...still, they need to do the research and wake up. It is their responsibility as citizens.