My wife is a VA RN, and also an Army vet (Iraq). Very few of her co-workers are vets, so it shouldn't matter that you don't have military experience. Depending on where you're working, you could have patients ranging from World War II through Iraq and Afghanistan eras. She loves her WWII patients -- they're a whole different breed than the more recent vets, and still tough as nails. Outside of the fact that VA probably has better pay and benefits than anyone else in the area, they also have some of the best equipment available (even better than the local major university teaching hospital!). Another advantage that they have is that patient care is not dependent on insurance reimbursement. Your patients won't get sent home until they're ready, not because their insurance payments run out. You also won't have to account for every bedpan or box of tissues.
On the down side, it is a government institution, and it can be bureaucratic at times. As a government entity, it is also hard to fire the lazy and the incompetent, so you may find that you're doing the work of two or three people, just because someone else feels that they're entitled to a paycheck for sitting on their rather ample gluts in the break room or the nurses' station. The quality of leadership can also be hit-or-miss.
Overall, it's been a very rewarding experience for my wife, and she wouldn't want to work anywhere else. She has found that the overall quality of nursing care, particularly from the RNs, is a cut well above what you would find in your average hospital. I hope you have a long and rewarding career with the VA, and take good care of our battle buddies!