-Returning teachers don't need to get it for contract renewal (unless their particular cram school demands it).
-it's not retroactive for job-seekers who entered before that aforementioned date in June
-If you're working at a public school or Uni, you don't need to bother with it. (Is that fair? Well, no. But as a Uni teacher, I'm glad I don't have to bother with this rubbish.)
"Good conduct certificate" is so vague though. I'm still not sure if they're asking for a local jurisdiction police check (which only take a week and are relatively easy, but also pretty useless for determining criminal background), or a full-on FBI check (which takes about 3 months non-expedited, or costs a crapload of cash if you want to expedite it).
EDIT: Ooooh, okay. According to the Taiwan news article, it is an FBI check. Those are a pain in the ass. Good luck, OP.