The following is paraphrased from the Saga JET guide: 

Sick leave, or byokyuu 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 runny nose, are hungover, or  want to go get some form of non-emergency check-up (sight test, teeth cleaning, etc.) then you will have to use paid leave (nenkyuu 年休) instead.  

To take sick leave, your process may differ if you are Municipal or Prefectural, but the general flow should be as follows:  

1. 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.  

2. Visit the clinic/hospital and receive a doctor’s note (shindansho 診断書) and receipt for any treatment/medication. You must ask for the doctor’s note as they will not automatically give it to you, and it is not free. This note alone can cost about 3,000 yen.  

3. 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.  

You may notice that most Japanese teachers will not use their byokyu when they are sick, and instead 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 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 and your contract when and how byokyu can be used, and the procedure to follow, before you get sick.  For more information, check the Health section of the JET Programme General Information Handbook.