Being a veteran doesn't change whether the loan is recourse or not. It matters mostly what state you are in (and perhaps the type of property financed). If it is a VA loan, the VA can help you if you are behind on your payments, see: https://www.benefits.va.gov/homeloans/
If not, I suggest you talk to the lender about other options (deed in lieu, short sale, etc...) rather than walking away.
Walking away will TRASH your credit rating in any circumstances. Whether there is any monetary issues further depends on whether the loan is recourse or not and how much equity there is in the property.