Possibly, if the patent is in your name only with the USPTO and you did not sign over the patent to your company, you would technically still have legal rights to the patent.
However, the inquiry as to whether you could maintain the rights in the patent lies in the contract language of the contract that you signed with your company with respect to a material breach of the contract and the remedies set forth within the contract. Your refusal to sign over the rights of the patent would most likely be considered a material breach.
The second inquiry would be whether the contract could be assigned to a third party such as this supplier who could enforce the contract against you by "standing in the shoes" of original party to the contract, your employer.
Thus, to answer your question, you or your attorney would need to review the contract that you signed with respect to those provisions. As of now, the fact that the supplier is contacting you directly without some type of "cease and desist" may indicate that they are not on strong legal footing with respect to the patent rights. Nevertheless, it would be a good idea to retain an intellectual property and/or contract attorney to represent your interests in the patent.
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