It's not an "Aspie thing," IMO. I am NT, married to sackcoat, but have always had issues using public restrooms. I use the "hover" method while using it, and I never directly sit on the toilet. I also have a tendency to hold it until I am either at home or our hotel, or my parent's house, or whatever, just so I don't have to actually use public restrooms. When opening the door, I ALWAYS use my sleeve or a paper towel. I also hardly ever wash my hands in public restrooms, unless the restroom seems relatively clean. I prefer to use hand sanitizer because the sinks are so dirty, it seems it would be counterproductive for me.
I should also mention that I have been the same way at my places of employment. I can probably count on one hand how many times I used the restroom at my job, even while I was pregnant. I guess you could say I have a big issue with public restrooms.