If you make it to the airport, you will get a stamp in your passport about the overstay, you will have to pay the fine, it will be registered in the database.
There is no reason or excuse to go into overstay and when you do, you face the consequenses.
You can apply during the last 45 days of your stay.
But be aware that if you go on the last day and do not have all the documents and have to come back the next day, you are in overstay. Better is to go a few days early as the 1 year Extension of Stay is added to your current stay and you do not "Loose" days by going early.
IF the hospital makes the letter for you and Immigration agrees you are unable to travel then you would get a new Extension of Stay based on medical reasons.
Removed advice and discussions about the Thailand pass and insurance. This page is called Thai Visa Advice, please keep your comments and advice to the point.
Where how what and how much for how long he can buy an insurance in or outside Thailand has nothing to do with his re-entry using his Extension of Stay.
If you re-enter Thailand you use the re-entry permit that you bought before travel at Immigration or at the airport to keep your currect Extension of Stay alive.
BUT you need an insurance for the Thailand pass, which has nothing to do with a Thai visa.