I don't even know that the contract has such a provision, let alone its precise language.
But it should have a provision that bars the employee and his agents from making disparaging comments about the employer. Note, the prohibition would not be limited to "false" statements, but would proscribe "disparaging" statements. Accordingly, a statement while arguably true ("redshirt" vs. "fifth-year") could still be disparaging and actionable.
Further, if the contract contains appropriate language, it not only should allow Notre Dame to terminate the employee without paying the otherwise due-and-owing severance compensation, but it also would be grounds for a breach-of-contract action by Notre Dame against the employee.