I use fastidious sometimes, usually to describe a person is very picky, perfectionistic, etc. It's kind of a funny sounding word, so I almost feel like I'm joking when I use it, but it's descriptive. Squeamish is a good word. People frequently use 'uncomfortable' to mean the same thing, though 'squeamish' is more precise. The usage is slightly different, "Squeamish about public nudity" or "Uncomfortable with public nudity."

Instead of fastidious, I would use meticulous or perfectionist. “He is super meticulous” or “He is a perfectionist.”

I am “squeamish” when I am uneasy or uncomfortable such as facing a live lobster or being on a roller coaster. For some it may mean feeling nauseated as from a boat ride.