I really wonder if there is a negative of shall. I've heard something like shan't. For example I shan't or Shall we go to the cinema? No, I shan't. I don't know whether this usage is correct or not.
The negative of shall is shall not. All modal verbs have contracted negative forms, and the contracted negative form of shall not is shan't. Shall and its negative both seem to be in decline and their use is limited to mainly to legal contexts and the expression of politeness in some questions.