I"m a germophobe so I know what you mean. Even if the janitor has just been in there, I'm still squeamish.
When I was little, my mom had a way of helping me balance over the toilet so I could squat without touching anything. It's good practice for your daughter to be able to master this skill for when she goes to a public bathroom without you. My mom would stand facing the toilet and stick one knee out towards me. I would wrap my arms around her knee and "hover" over the toilet. It works great!
My son is 3 and he asks me to hold him "airplane-style" over the toilet for now. I'm not looking forward to potty training my daughter!
Also, to the person who found that drinking fountains are nastier than toilets - did you check door handles too? I remember doing that experiment and then not wanting to open doors for the longest time. Yuck!