You can have a veterans designation printed on your Nebraska driver's license or ID card as proof of service. Adding it when you renew is free; adding it before then means paying the regular replacement document fee.
Register first on the Nebraska Department of Veterans' Affairs Military and Veterans' Registry. Once NDVA confirms your registration, ask for the designation at your next renewal for no extra fee, or apply for a replacement document online or at any driver licensing office and pay the regular replacement fee.
Your DD214 or other discharge paperwork to complete the Military and Veterans' Registry entry, and your current Nebraska license or ID.