This again?! 
Okay, here’s a brief introduction to the topic:
Tl/dr: This area of law just isn’t that reliable in Taiwan, at least not since the Nanshan Insurance fiasco.
The first argument (he taught more than one class ergo he’s an employee) would easily be rejected, though it might carry some weight the other way around (he only taught one class ergo he’s not an employee). The other arguments are sound, and I would say they should be enough, but… this is Taiwan.
Unless something has changed in the last few years, this is almost but not quite true. It doesn’t make a huge amount of sense, but the logic goes like this: a buxiban can be established by a natural person or a legal person (usually a company), but even if it’s established by a legal person, the buxiban itself is a separate business entity with its own (de facto) legal personality – technically not a person, but still capable of being a party in its own right – so when you contract with or sue the buxiban, you’re not contracting with or suing the company.
What this means for labor insurance is that the buxiban is not subject to the five employees rule, because that rule is only for companies and “行號” if I remember the wording correctly – “firms”. The choice of wording was probably intended to be broad enough to capture most types of business entities, but someone in the government decided to issue an official interpretation that said it doesn’t include buxibans.
However, if the buxiban has already established a labor insurance unit, then insurance is mandatory for all employees, full- and part-time. Even if no-one is registered at the moment because they only established it for one person’s sake and that person has left, the mandatory nature of the insurance persists. (Incidentally, failure to register an employee for labor insurance has been found to be sufficient to trigger LSA Art. 14 in case you want to quit with no penalty, though this being Taiwan, precedents are usually non-binding.)
Like I said, this may have changed recently. If things are getting serious, talk to a lawyer… and then get a second lawyer’s opinion. 