First, I am not a lawyer.
No-hire and/or non-solicitation clauses can be added to your contract (MSA/PSA). My understanding is they are variably enforceable in different states.
They are generally not enforceable against employees (you are better off developing and maintaining a good relationships with your employees, then trying to force them to do something).
That said, your agreement with your business customers is generally a binding agreement, and should they try to hire your people out from under you, you should have grounds for a lawsuit. Suing your customers has other implications that you may want to consider.
I had a situation where a customer tried to hire one of my employees who had been working on his project for a couple years by that point. He was trying to reduce his expenditures. My employee notified me that this was happening, because we had a good relationship, and I contacted the customer and reminded them of our contract agreements.
This is considered an ethical breach. We weren't providing contract-to-hire headhunter services. Most people who hire vendors understand the distinction, and this customer knew what they were doing (they were being intentionally sneaky about it). At the completion of the contract I fired them as a customer, and moved the employee to another project.