I'm an anarchist/existentialist by philosophy. In general, I dislike all top down governments, though I think there are some that are better than others.
I think a fundamental point here is that people should think about the situation globally, more than just about themselves. There are ~300 million people in this country and ~7 billion people in the world, and we're all humans. Where are the values of compassion, integrity, and sense of existential responsibility for our fellow humans. Or is there no value in thinking about such values? Even if there isn't, and it's a pure pragmatic issue, you're not alone on this planet and there's an "average" desire that triumphs your own by sheer force. One can go on about capitalism and Ayn Rand and lifting yourself from the bootstraps, but if the mob decides to take away your millions, they will, and there's nothing to stop them if you and they don't share the basic values described above.
Specifically, regarding mandatory health insurance, the question is whether you believe health care should be a basic provision for all people like police and firefighters, or is it every person for himself? If the latter, why stop at police and firefighters? If the former, then you need to accept there's a cost to providing healthcare for everyone (including you) and we're all collectively taking care of each other. So you may be healthy but your kids might end up being born with a debilitating disease and you might not be able to afford to pay their health care, but I am happy to subsidise you with a portion of my income because I realise it's only a matter of chance that I'm not in your position. Sure, I might subsidise some people who don't deserve it, but I'm happy to accept that to help out your sick kid. That just means the method by which the taxpayer money is distributed needs to be scrutinised, not the basic philosophy of helping those in need.
I have no problem with a progressive tax where I nonlinearly pay more as I make more. I do so. When I first did tax returns where a surprising amount of it went to taxes (after I graduated from all my studies), my tax preparer said "that's the price for making a lot money". And it wasn't a big price to pay. I still got to keep a lot of it compared to other countries and I'd happily pay what I pay in taxes just for the National Park System alone (that has given me more happiness than the spending on the military ever will).
Again, my problem with the Ayn Rand idea is that you're not alone on this planet. Everyone wants something. We all have to live together and do so in a happy manner. You could be the most successful person in the world but if there are no roads, no police, no firefighters, no health care, even if you don't think that's an existentially dubious position to be in, you'll have a mess of people outside who will break your head and take what you have from you.
And if you really want to see what limited government in the form of "let's drown the baby in the bathtub" Gingrich model advocated by the Teapartiers is like, go to Somalia. There it's every person for themselves and it's all one big "private" enterprise. Being a warlord happens to the best kind of business to be in. Judging by all the gun and militia I talk I hear, people would like to go "back" to those wild west days. I love westerns and I really romanticise them and would love to play Lucky Luke, but in all honesty, I'd rather be secure and healthy and happy with my two year old daughter without bullets flying around.
Some of the biggest personal threats to our lifestyles are indeed assaults against freedom of speech, expression, and privacy, particularly in the Internet era. But these assaults are carried by government and industry alike. It's important to not confuse communatarianism ("do unto others" and its contrapositive which include things like taxes to pay for cleaner air, police, firefighters, education, healthcare, security for your family if you die early or get hurt, or when you grow old) with greed for power. The latter is not unique to government but is also one of the worst aspects of capitalism and humanity in general. When government or industry do the former, it is good. When it is not, it is time try again, but to achieve the same ideals.