Your friend is right to be concerned generally about public restrooms, in that there are some infections that can get picked up from coming into contact with any surface that is contaminated with feces. The types of conditions that could be theoretically picked up would be things like stomach bugs and diarrhea
, on the other hand, are not an infection, so that would not be possible.
Most of the time, the risk of getting an infection from using a public restroom is low, especially if the restroom is clean. Of course, you should avoid using a toilet that is visibly dirty. You can also consider using the toilet seat covers that most restrooms supply. Finally, the most important step is to make sure you wash your hands really well when you are done; alternatively, carry an alcohol hand cleanser with you and use that.
If at any point you or your friend are concerned that you may have picked something up from a restroom, the best step would be to make an appointment with your primary care doctor
, who will be able to check things out for you. Also, they can give you advice about good hygiene and how to keep yourself safe in public spaces.