What this is first and foremost is trademark infringement for using your logo without your permission and you need to send the student a letter telling them to cease and desist and you could also sue the student for the trademark infringement as well as for any known false statements they have made.
Next, you merely need a simple contractual agreement stating that "student/employee understands and agrees that they have no permission to use the logo of the school in any manner and for any purpose without express written permission. Furthermore, student/employee understands and agrees not to make any disparaging or derrogatory remarks about the school to any third parties nor publish any such remarks. The student/employee further understands and agrees that any violations of these provisions will result in legal action and the student/employee agrees that if legal action is brought they shall be liable to pay any and all legal costs, including but not limited to attorney's fees for the school pursuing such an action in breach of this agreement. Additionally, any such violation of this agreement shall be grounds for immediate dismissal from the school without any refund and/or termination of employment misconduct directly related to employment such that I understand unemployment benefits should be denied." Then have them sign the agreement.
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