If you have a normal work permit through a buxiban, the buxiban must (contractually) give you at least 6 hours of work (including lesson prep) per week if it’s your second, third or fourth work permit (while holding a permit with a main buxiban employer), or at least 14 hours per week if it’s your first one (the main one).
And if you have a normal buxiban work permit, it’s very likely that your employer has filed a normal-looking buxiban teacher contract with the Workforce Development Agency in order to obtain that work permit, and a normal buxiban teacher contract makes you a worker within the meaning of the Labor Standards Act and the Labor Insurance Act, ergo the buxiban must register you for labor insurance if it is already a labor insurance unit, which it is if it has any other workers registered for labor insurance or ever has had any (or has established a labor insurance unit even though it hasn’t gotten around to registering any workers).
If you have a normal buxiban teacher contract but work fewer than 6 hours per week because as an APRC holder you do not need a normal work permit, you are still legally a worker employed by the buxiban.
It is possible for a buxiban to hire you as a mandatary (independent contractor) with a so-called commission contract, but if push comes to shove, both parties will have headaches trying to prove what type of contract it really is, regardless of the title of the document.