Anyone has a right to apply anywhere, to any school they want. And admissions committees have the right to reject applicants, which is likely to happen if the extent of the issue here (requiring hospitalization and lengthy, continuing homebound schooling) is revealed. That is why I advise the applicant to arrange that it NOT be revealed, if at all possible.
And yes, people with serious, unstable medical or psychiatric issues are well-advised to stay close to family support, for college.