Starting outside the fall term is usually not a problem for undergraduate programs (after all, you can change majors at pretty much anytime during your college career), you just might be out of sequence and end up staying an extra semester because of the way classes are sequenced - that's more likely in specialized majors such as engineering.
Graduate schools can have restrictions on when you can enter, many only allowing you to start the graduate program in the fall - that might be on a departmental basis rather than a university basis though.