Though I prefer the term 'classical liberal,' I'm a libertarian. I am highly skeptical of government, both in intent and general competence. I wish the government, and the federal government in particular, did a lot less than it is currently charged with. All that being said, a federal government shutdown will likely be a disaster and a mindless exercise of political self-mutilation.
For one, it won't stop ObamaCare or any other entitlement or longstanding government program. Thus, all this is about being obstinate for its own sake. It won't be the end of the world if it's shut down for a few days, but what will happen is that government services upon which people depend, rely, or otherwise enjoy will be unavailable to much of the public or otherwise crippled in their effectiveness. And who will get the blame for this absence/dramatic reduction in service? Tea party/libertarian/small government types because the people responsible pay hollow lipservice to their ideals.
This is the wrong way to sell smaller government.
Think of it this way: take an addicted cigarette smoker. You want to convince him to quit for his own good, but he's not ready. Well, instead of reasoning with him or convincing him there are alternatives to smoking--nicotine gums, patches, e-cigs, jogging, whatever--you just steal his cigarettes and prevent him from getting his fix by whatever means you can.
Now this guy is going to be pissed at you. This is because you haven't convinced him he doesn't really want that cigarette--you have just taken it from him and made him angry. Very, very angry. Even though smokers know the stuff is bad for them, many who may want to quit haven't found a way that works for them. You've now made that decision for them, but you know it's only a temporary one.
But as soon as you give this guy access to cigarettes again, he'll go buy another pack and may even smoke more. You haven't convinced him there are concrete steps he can take to get over his addiction--you just convinced him you're a stubborn prig who'll make other people miserable just to prove a point. You, and others perceived to be like you, will likely have much less sway with him from that point on because you were an unthinking, uncaring, self-righteous S.O.B.
The GOP is that S.O.B.
bellum medicamenti delenda est