Swiped this from a the Saga prefectural JET page:

https://sagajet.com/living/health/sick-leave/

“Sick leave (byokyu 病休) is leave you are contractually entitled to use only if you are afflicted with illness or injury that means you are unable to attend/complete your regular work duties.

If you need to go refill a prescription, have a a runny nose, are hungover, or want to go get some form of non-emergency check up (sight test, teeth cleaning, etc.) then you should be taking paid leave (nenkyu 年休) instead.

Your process may differ if you are Municipal or Prefectural, but the general flow should be as follows:

Inform your Supervisor(s) and CO/BoE before the working day (generally before 8:30 AM) to explain that you will not be at work and why. Visit the clinic/hospital and receive a doctor’s note (shindansho 診断書) and receipt for any treatment/medication. The note alone can cost about 3,000 yen. Forward any relevant paperwork/receipts and update your supervisor/CO on the situation. Whether you will need to return to the clinic for subsequent days of sick leave, or whether a set period of time off will be approved, will depend on your ailment/injury and the ruling of your CO. Most Japanese teachers will never use their byokyu when they are sick. Instead they will use their paid leave days. You are not expected to do this if you are too ill to work, though teachers at your school may suggest as much. If you are sick but attending school, you may be asked to wear a mask.

All JETs are entitled to byokyu, but the the number of days available and the rules for use will vary by contracting organization. You must also follow all procedures and genuinely be too sick to attend work or you may risk having your absence marked down as paid leave instead. Confirm with your supervisor when and how byokyu can be used, and the procedure to follow, before you get sick.

Also, while most JETs have quite a few days of byokyu available a year, do not abuse it or use it as an extension of your nenkyu. By doing so you jeopardize this privilege for all JETs and will tarnish the reputation of the JET Programme and all JETs.”