In my experience, the process started with the offer/contract (about 6 months later, I got out of quarantine).
After the offer/contract, the first major document task was getting my highest completed degree authenticated by the TECO office closest to the issuing university. The employer needed the authenticated document to apply for my work permit, for which they also needed a copy of my passport and an electronic Taiwan-sized passport photo. If they haven’t asked for these things, they probably aren’t ready to approach the government about you yet.
The work permit was sent to me so that I could apply for the visa. The TECO office where I applied for the visa said they needed a health form, but actually they did not want it when I tried to give it to them with my complete visa application package (with police check, etc.).
So, I would suggest you read “complete this health form to start your application for a visa” as “complete this health form, so that we know you are committed to joining us and healthy enough to fulfil this commitment, before we make you an offer (let alone start your work permit application which will be required down the road for a visa)”.
Once they have submitted your work permit application to the Ministry of Education, since it is tied to your passport number, it seems to me that then applying for another work permit with a different company could cause you complications.
Why not wait until you are ready to sign a contract, because you have made up your mind, before you worry about the paperwork?
That said, the TECO authentication of your degree will be required anyways and isn’t tied to anything and doesn’t expire, so you could start there…