When you tackle a certain problem and try to apply a libertarian framework to it, odds are you have to check if it's an implicit monopoly that you're dealing with and go from there.
In your particular example you're dealing with a city government monopoly for health inspection. It's not free and is paid by citizens in the form of taxes. If there was no government and citizens believed it is in their interest to have such an inspection, surely you'd have at least one, but possibly two or more competing with each other and financed by consumers voluntarily. When you have a government, what you have is a monopoly which is, apart from all other evils of monopoly, much more vulnerable to corruption. If a government official is bribed and it is revealed, he may go to jail, but the agency itself stays in business. Which means it has almost no incentive whatsoever to prevent corruption among its employees.
Finally, government doesn't really ask people if they need such an inspection after all and if yes, how thorough the inspectors should be: maybe businesses in this town are exceptionally honest, or maybe, on the other hand, they are exceptionally sloppy. So what you have with a government in place is a monopoly which business operations are based not on the actual demand for its services, but rather on some metrics that government officials came up with, which may or may not be useful.
He explained this pretty well. How come most ecommerce websites have "trust badges"? The government didn't need to create an agency to rate web site security, but yet such a thing exists.
If people wanted a health inspection organization to rate restaurants it'd exist. Since the govenrment already monopolized that field it doesnt. Okay, it sort of does, it's called Yelp.
Anyways, the point is pretty clear, we have all sorts of "trust" systems on the Internet to verify everything from privacy (Truste) to security (Verisign, etc.). People can create these types of organizations on their own without government and will if they actually do something useful.
My point was, 5 years ago no one imagined a concept like Square would exist. Just because you can't imagine a nongovernmental organization that serves the same purpose as local health inspectors doesn't mean it's impossible to have such a thing.
On the contrary, quite a few similar organizations do exist, but only when a government monopoly doesn't kill competition in that space. For example, who in the 90s would have guessed that something like TrustE would exist. It's not that far fetched to imagine that if government wasn't policing restaurants a private organization could exist to provide such a service. Yelp doesn't do that because it's not their mission and no one is going to make that their mission since the government already monopolized that job.
I'm not saying we should have 0 government either, but if we're not willing to question the necessity of things as trivial as local government health inspectors then where do we draw the line?