The primary uses of can are to express possibility and ability, with permission a third. Will expresses chiefly prediction and secondly volition. Should expresses mainly obligation, secondarily deduction and thirdly unreal and tentative meanings.
In your first example, the speaker uses can’t to express inability to continue in the relationship. We don’t know the reasons behind this inability, but they are clearly of an overriding, and presumably emotional, nature. Had the speaker said I won’t continue in this relationship we would have gained the impression that the emotional wherewithal was present but that other reasons had prevented the continuation of the hitherto happy state of affairs. In practice, I won’t continue in this relationship is likely to be followed by a clause beginning with, for example, unless . . .
If a speaker says You can't embarrass me in public, the implication is that the person addressed is unable to embarrass the speaker in public without certain consequences ensuing.