It's not a student loan. Not even close.
If the former employer goes to a new employer and says, "I have a contract with your employee that says I can 'garnish' a huge amount of money out of their paycheck", the new employer is going to... not care. They are not going to "garnish" the employee's wages. Even if the employee consented, I doubt that they would cooperate with the arrangement as opposed to telling the employee to make the payments directly.
For anything else, the employer seeking reimbursement would need a court order for garnishment and, no matter what the contract says, they're not going to get a court to issue an order of wage garnishment for amounts in excess of what is permitted by state and federal law.

