He is not free to lie to a competitor's clients in order to try to take away the competitor's business -- that can rise to the level of tortious interference with contract.
Quoting Prudential Ins. Co. of Am. v. Fin. Review Servs., Inc., 29 S.W.3d 74 (Tex. 2000).
Absent additional facts, I'm not willing to join that generalization.Quoting cbg
You don't need a written contract to protect trade secrets -- an employee who steals trade secrets can be successfully sued even if he has not signed anything with the former employer.
However, in the absence of an agreement, the employer must establish that the customer list was a trade secret,Quoting Trilogy Software, Inc. v. Callidus Software, Inc., 143 S.W.3d 452, 463 (Tex.App.-Austin 2004)
For an obvious example of a customer list that is not a trade secret, take a look at a typical building supply company's website and you will often find a search feature for its distributors and retailers.Quoting Numed, Inc. v. McNutt, 724 S.W.2d 432 (Tex. App.-Fort Worth 1987, no writ)
It's not necessary that a misappropriated customer list be an actual document or electronic file, but it must actually qualify as a trade secret before a former employer can successfully sue over its misappropriation.
As the former employer under discussion is making false claims of having a non-compete, not having his lawyer send a cease-and-desist letter based upon misappropriation of trade secrets, there's a good chance that the customer list is not a trade secret. (It could also be that the former employer doesn't understand his actual legal rights.)

