The answer is no. But that does not mean you should not consider consulting an attorney and getting a separate non-solicitation and non-disclosure agreement from them.
Remember that your employee handbook is not an agreement. It is a set of policies that are enforceable without any agreement between the employer and the employee. It is unilateral. For example, employees need not agree to be on time in order for the employer to be able to fire them for not following that, or any other legal and well-constructed policy. Also, note that policies govern the behavior of the employees and set standards while at work. We are very careful to write policies so as to not have them misconstrued as agreement language by the way. It takes a ton of education and experience to be good at it.
Once an employee separates from employment, a handbook policy pretty much has no teeth, which also means it is not enforceable. Therefore, in order to be effective and not a liability while also binding them to not solicit after employment ends, employees must sign an enforceable agreement. It is in fact a contract and should be separated from your policies and signed separately.
There are other problems, beyond enforceability, with creating defacto agreements inside policy handbooks. It can cause serious issues and negate all the good things within your entire employee handbook. Here is how:
By misplacing “agreement” language within your policy book you are opening a very wide door to the argument that all your policies are no longer policies, rather they are all binding agreements between the parties. The argument is that your handbook is now a contract! Remember, policies are unique within US law and give all the power to the employer. One of many issues is that once you inadvertently turn your policy handbook into an agreement, anytime you want to make any changes whatsoever, you must negotiate with the parties to the agreement and give them something for agreeing to the changes. That would be no bueno. If we want to change work hours, add a dress code, alter the amount of time off available, or any other of a number of things we need to be able to do so unilaterally as a matter of law.
Another reason you should not include a non-solicit in your handbook is because in many states they are not enforceable with lower level employees. Depending on your state, there may even be a salary limit below which a non-solicit is unenforceable. For this reason, non-solicits are generally only appropriate for your office manager, associate doctor or a hygienist in some cases.
Non-solicitation agreements are legal in almost all states. They must be carefully and narrowly crafted so as to not violate state laws while also being enforceable. They should never be included as a policy. They should be kept separate from your employee handbook.