I don't have any firsthand knowledge of your specific situation, and I'm not licensed to practice law here, so this is definitely not legal advice. I'm just pasting text here that appears to me as if it might apply to your situation. The stuff below is essentially the same stuff that I've pasted elsewhere on the board, but I've modified it a little:
From the [color=#000080]Employment Services Act[/color], in pertinent part:
wda.gov.tw/en/home.jsp?page ... 1304190012
From "[color=#000080][b]Qualifications and Criteria Standards[/b][/color] for foreigners undertaking the jobs specified under Article [color=#000080]46.1.1[/color] to 46.1.6 of the Employment Service Act," in pertinent part:
law.moj.gov.tw/eng/LawClass/LawA ... e=N0090031
Again, from the Qualifications and Criteria Standards, the same document as the one quoted immediately above:
From Examinations for Professional and Technical Personnel in Taiwan(ROC):
wwwc.moex.gov.tw/english/content ... enu_id=509
headhonchoII is experienced in the medical and scientific fields, so he might have more information about the employment situation in those fields, if you can get it out of him.
yyy, Feiren, Omniloquacious, and some others may have legal and other kinds of information.
I hope this helps, or at least does no harm.