In the 70’s they came up with a swine flu vaccine. I was in the Navy at that time. My dad worked for a company that had many microbiologist on the staff. The prevailing opinion there was to not get the vaccine so my dad told me to refuse it. I did. I was repeatedly ordered to roll up my sleeve which I refused. I told my command they would have to discharge me however they wanted to but I was not and did not take that vaccine. Soon after the program was canceled due to people getting some syndrome. I was never court martialed and stayed in the navy. So service members can refuse the vaccine and be discharged in effect quitting.
You got lucky, in the ,1990s the CID went after anthrax refusals with a passion. Every one I had access too has a BCD at minimum quite a few were dishonorable when they refused a second time and demanded a full court review well they got what they asked for.