“especially attorneys and legal scholars discuss whether the suspended cadets can fight their punishments on free speech grounds considering the Citadel is a state funded public institution?”
Sure they can. But I don’t know enough about the law in that circuit to say how good their chances are. If their names are not currently public, it might be unwise to sue, even if they have a claim. Speaking from experience, I’d say they have a reasonable chance of being able to transfer to comparable colleges if their academic records are OK. But having their names attached to that image forever would be bad news.