Can't really answer that one for you, but some things to consider are this. She probably should not sign the agreement not to compete if she intends to compete, even though arguably the non-compete may not be enforceable since it is forced as an adhesion (sign it or get fired) contract.. She may not want to resign as she may lose her rights to unemployment income that she may have if she is terminated. If she is working under contract, then the contract and its terms are binding. They can't force additional terms if she is working under contract. She should find a local business attorney to speak with concerning this. This attorney can probably assist her in setting up her new business and devising a good business plan.
This answer expresses only general statements about bankruptcy and/or debt defense and does not constitute advice to you in any form and does not create an attorney client relationship between the party asking the question and the party providing the answer. Furthermore, all cases are fact specific and it is not possible to give you legal advice without a complete evaluation of your case or if you are already represented by an attorney. Therefore, this answer does not constitute legal advice of what you should do specifically in your case. You should seek local counsel to help you with your specific issues.
I agree with David. In addition, your question references your wife's clients, but I would caution that the clients are most likely not your wife's clients, but clients of the company. Information related to those clients is likely the company's property, as well.