What are the regs on quarters slip?

Army Regulation 40–66. I'm not finding anything in that regulation stating Quarters has to be given by a military medical facility. However, I recommend you contact Tripler and speak with someone there that may be able to answer that best.

Communication will be your biggest issue or saving grace when it comes to things like this. You need to understand that the Command prefers documentation from local military treatment facilities. The nearest ER to your house may or may not fit that bill which is what seems to be creating static. You just need to followup with your local provider and get concurrence through them. It will provide up to date information for them and your Command. Good Luck.

Part of the problem is all the soldiers before you that tried to work the system. On this note I do not see a facility address you were seen at or a provider's name. To me this looks like you could have typed it up yourself (I am not saying you did) and is nothing I have seen any medical facility give for years. All my notes I give patients my name is legible and has my licensure as well. I have not seen an active duty medical provider give a note in years without a stamp. You need to get yourself to a military facility and get paperwork from there and all of these issues will go away.