Consult with a patent/trademark attorney in terms of how you can change your product so you can publish it without being liable for misappropriation. Non-compete clauses should be signed voluntarily and no one can force you to sign them.
It's very likely that the creation you refer to was a "work made for hire" and, therefore, the property of your company, regardless of patent. Any sale or publishing of this information could be considered misappropriation of confidential information or trade secrets and subject you to claims for damages, despite the fact that you have not signed a noncompete. You can of course refuse to sign any agreement you like, particularly on termination of employment, but for the above reasons, I don't think it matters vis-a-vis this software. That said, you should have an in person consult with an attorney with expertise in intellectual property issues before you proceed, certainly worth a more in depth review.
If you plan to start a competing business, I strongly recommend getting legal advice about your responsibilities as a former employee. It has nothing to do with whether you had a valid noncompete agreement or not. You likely have additional fiduciary responsibilities as an employee that may prohibit you from using material, nonpublic information at your new company. Otherwise, you risk liability when your former employer discovers what happened. The benefit of successfully launching your company far outweighs the minimal cost of getting good legal advice before doing so. Be sure to speak with a New York lawyer about your rights, as employment law differs in each state.
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