Contraction at the start of the sentence. Can't/Shouldn't etc?

Can you not? Should you not?  You may also substitute 'you' for 'one'.

"Can you not do it?"

Contractions are not "informal" and are used in formal speech and writing. 

Breaking them up sometimes changes the meaning."Can't we be friends?" is an offer of friendship.If you make it "Can we not be friends?" it sounds like you want to end your friendship."Shouldn't we tell him?" suggests that telling him is the right thing to do.
"Should we not tell him?" suggests that we are not sure which is the right thing to do.

"Aren't" "don't" and "isn't" can be broken up, if you want to, without changing the meaning, but it's usually done for emphasis or emotional effect. "Are we not free citizens?" "Do we not want him to succeed?" 

Possibilities include Can you not do it or Should you not do it.

Can't you do it? - Can you not do it?

Shouldn't you do it? - Should you not do it?

The second one is a bit odd. Strictly, it means you shouldn't do it.  However, if I heard it in conversational speech I would interpret as:

You should do it.