You still aren't criticizing the argument. You're criticizing the motivations of the people who wrote it.

Yes, I am. The flaw in the argument is a basic type-error: Type MORAL does not match type FACTUAL. You cannot use a moral claim (government is evil, taxation is theft, minimum wage is interference in free contract) to justify a factual claim (minimum wage increases unemployment).

The data shows that there is a correlation between the minimum wage level and unemployment among lower-educated workers. That doesn't prove that the theory is correct, but it supports it pretty well.

First of all, which data? Second, most major studies have found that minimum wage has little to no relation to total unemployment, particularly not once we factor out cyclical trends. Unemployment is mainly governed by the cyclical and structural trends of the economy as a whole.

Now, at the margin, does the minimum wage disemploy some people? Probably, same as the 40-hour work-week does. However, I've got two counterarguments here:

A) Any wage below a living wage (enough for the worker to support themselves on the wage alone, not necessarily anyone else) is either slowly murdering the employee (who cares, we're not moralizing, right?) or taking an implicit subsidy in the form of government top-ups to that employee's income (what actually happens). This leads right into:

B) The wage-level that a given business can support at a given level of worker skills is determined by many different variables about capital investment. For example, today's roboticized factories have a very hard time employing low-skilled and mid-skilled labor, while yesteryear's assembly lines had an easy time of it. At the very bottom, you need some capital-intensivity to support anything about subsistence farming, but there's loads of what are basically "user interface issues" determining how much capital-intensivity you can support for how little user skill. This is something us technologists can Work On.

This leads to related point:

C) Implicit subsidies for the kind of capital-unintensive, unproductive, sub-poverty-wage work being done in sectors like retail distort the market, both by suppressing the natural organization of labor (collective bargaining was the original response to sub-poverty wages when there was no minimum wage) and by padding the profit margins of businesses that use labor to substitute for capital investment (ie: in a market without income top-offs, I believe Amazon would more quickly eat Wal-Mart's lunch, precisely because it has automated away more of the retail process).

D) Many companies pay more than $9 or $10 per-hour for low-skilled labor while their competitors pay less, so what's stopping everyone from doing so?

So the total conclusion is that while the minimum wage should theoretically have some disemploying effect, and some studies have found a weak relationship when zooming in on the lowest-skill workers, these effects appear to largely "wash out" of the economy as a whole due to other factors that affect things more strongly: overall demand, overall demand for labor, labor organization, ordinary business cycles and capital-intensivity at set skill levels.