A few days back, the Atheist Ethicist spent some time arguing against public health care. He criticized it from a number of angles, principally that it is a private benefit (with a few exceptions) rather than public and that it's a subsidy for poor lifestyle choices.
The second is simpler to address: take smoking. It's a choice and it's bad for you, full stop. Why should the public at large pay to address the poor health brought on by your poor choice? This is a compelling argument that ignores the fact that the majority of health problems are not brought on by poor choices, but are the result of illness, accidental injury, or poverty. I deliberately avoided his example of obesity because when you're poor, the cheapest food choice is often going to lead to an ironic combination of obesity and malnutrition. Even in the case of self-inflicted illness as with smoking-induced lung disease or a heart condition from years of being overweight, I'm willing to argue that treating the health of the public is a public good and for the exact same reason that education is.
The ethicist is more than happy to grant that items like national defense, police and fire protection, education, etc are all public goods. Why exactly is education a public good, though? After a few centuries of demonstrating, in America and Europe, that free education is for the benefit of all, it should seem obvious, but let's go ahead and explore it.
Imagine a person born into the world circa 15,000 years ago. He lives in a reasonably fecund area of the planet and, although he's five or seven thousand years too early for agriculture, his nutritional needs are well met by a settled, hunter-gatherer lifestyle. He has some free time. Unfortunately, he's some ten thousand years too early for written music as well. He'd have made a great composer. Now imagine a person born in Somalia today. What are the odds he'll become a composer?
Education is a public good because it opens up the possibility for us all to achieve our potential, our dreams (with hard work and luck). Without it, we're lucky to scratch out a living doing whatever scut work is available. By giving everyone a solid grounding in math, science, literature, art, history, writing, music, all of it, we open them up to the possibilities and let them discover natural talents that, were they to languish in ignorance, they could never learn they had except by accident. We all benefit when we can all dive into our passions, master them, and create and discover wonderful new things. Minutiae of law, science, music, and art, even if you never hear about them, enrich your world and make it a better place. And none of that is possible without education.
So why is healthcare a public good? A pianist can't play with broken fingers. A diva can't sing with bronchitis. A janitor can't sweep on broken legs. We all benefit when everyone can explore their passions and their talents. Just as they can only explore them when they have the educational foundation to do so, they can only explore them when they have the health to do so.
Just as we should do what we can to keep our schools and our police well funded, well maintained, and well regulated, so should our public health, and for all the same reasons. When we're all free, healthy, and happy, we can do amazing things.