No, you don't have to tell the schools about your diagnosis. However, some schools have assistance programs to help people with disorders and disabilities. If you want such help, then you would have to tell them to get the help.
As long as you qualify for a scholarship, it shouldn't affect your getting one, as they are not allowed to discriminate. For the same reason, the schools can't refuse you just on the basis of having a disorder or disability. Only if it causes problems, like being too aggressive, or obsessively "stalking" someone, even if you don't mean them harm, or behave in other ways that interfere with the running of the school, the classes, and the other students getting an education. If you frequently have really bad meltdowns in class, this could interfere with the class and with others getting an education. However, it has to be something like these things, not just regular spectrum issues, to prevent you from staying at a college.
If you don't need any special services from the school, I wouldn't tell them about being on the spectrum. Instead, if people ask about any strange behaviors of yours, just say things like "I wear sunglasses inside because my eyes are light sensitive", or "I am sensitive to loud noises, so I wear head phones to moderate the sound levels." In other words, explain the quirk, without telling about being on the spectrum. Too many people either don't know anything about spectrum disorders, or what they do know is mostly wrong, so best not to reveal that you are on the spectrum, unless you need to access the special services provided by the school.