Do they have any legal right to my product?

Your question is really too complicated to fully answer on a site like this. First of all, you should see a patent attorney to learn if you invention is patentable, and if so, to have a patent application prepared and filed before you lose your rights simply due to inaction.  As to the your specific question, is inventing within your job description or would you be expected to come up with ideas such as the current one in the normal performance of your duties. If so, you company may have a claim for at least a royalty free license to practice the invention even if they have no right to own it. On the other hand, if this is not the case and you used your own time, facilities and money to develop the invention, it may be totally yours with your company having no rights.  Importantly, the rules governing such situations varies from state to state with some states providing greater rights to employees than others.  I do not practice in your state so that you should consult with a patent attorney there who can fully assess your situation and advise you more fully than I can.