Yes, you can ask questions in the negative. You can do this especially when a situation is different than you expect it to be. Why are you at the movies? Don't you have to work today? Why are you up so late? Shouldn't you be sleeping? That is very common. But your example is awkward. Also, it is common to use contractions: do not = don't Do not you work today? <-- awkward Don't you have to work today? Shouldn't you be working today? You don't have to work today?

Aren't you working today? Didn't you work today.

People do say: 'You're not working today?" or "You're off today?"

The use of negative questions is rare, however the use of them is still applicable today. for instance there's "do you not understand the question?" or as use in a request "can you not use that today". The use of negative's can be used within normal conversation there are no concerns when using that sentence.

"Do not you work today" is incorrect. Remember, to make a question, you need to switch the positions of the subject and the first verb. So "You do not work today" becomes "Do you not work today?", or "Don't you work today?" ("don't" is considered one word). It's much more natural with the contraction. Ex. (examples with * are synomynous with the previous example): You (do) live here. => Do you live here? You do not live here. => Do you not live here? *You don't live here. => Don't you live here We should do this. => Should we do this? We should not do this => Should we not do this? *We shouldn't do this. => Shouldn't we do this? He (does play) plays squash. => Does he play squash? He does not play squash. => Does he not play squash? *He doesn't play squash. => Doesn't he play squash? They are tall. => Are they tall? They are not tall. => Are they not tall? *They aren't tall. => Aren't they tall. You have been to Australia. => Have you been to Australia? You have not been to Australia. => Have you not been to Australia? *You haven't been to Australia. => Haven't you been to Australia? The abbrevated versions are usually much more natural, but people still sometimes use the unnabreviated versions.