They can unfortunately and many do depending on where the school is located. The public schools that both my son's attended did not allow uniforms to be worn when visiting the campus, even recruiters were told they had to wear civilian clothes. I remember once that a General was coming to speak to the class and he was told he couldn't wear his uniform, he declined the invitation.
The recruiters could not put up any posters in the career counseling office and when they would leave flyers the counselors would put a sheet of paper over them so nobody would get offended by seeing a person in a uniform.....I'm not kidding. When the district had a College Fair they put the branches of the military out of the main building under a stairwell far out of sight of everything.
Now the interesting part, my son's high school had the highest percentage of SA appointments and ROTC Scholarship winners in the district, a few times in the state. One year every UNSA appointment from our state was from their school. They also had a large number that enlisted in various branches as well, the CG always did good at the school. Even with the best efforts of the school to shield these poor kids from the military, those that were interested found a way, the advisors not sharing the schools approach helped a lot. When graduation came around they would list the appointments and scholarships in the program, I got a kick out of some people that were shocked that so many were going to SA's and ROTC Scholarships, I remember one woman saying "How could the school let this happen" Makes you roll your eyes sometimes.
This is the main reason they never had any presentation at any awards night from the military, those on the receiving end didn't mind, everyone they cared about knew and supported them.