Although they have a similar meaning, there is a slightly different nuance here. He said you would be back. -> Implies the person came back freely. He said you should be back. -> Implies the person came back because of an obligation. Another example for you is: He would not want your juice. -> Implies that the person would not choose the juice because he doesn't like it, out of choice - i.e. He doesn't like the juice, the juice might be dirty, he might not want to share someone else's juice. He should not want your juice. -> Implies that he may want the juice but there's something that is holding him back from freely choosing, like an obligation i.e. diet, allergies, was taught not to want it / accept it etc.