You DO NOT want to give any impression that you are an EMS vehicle. None. Laws very from state to state, but inevitably you are going to run in to a small-town cop who will take offense to impersonating an EMS vehicle and he will hit you with every violation he can find. Keep in mind EMS folks keep a tight knit group. The cop is going to treat you like you are impersonating a LEO, it's not going to be a good time.
It's just not worth it, and as others said, it's very unlikely you will be left undisturbed if a park ranger finds your vehicle. If anything, they'll think it's either stolen and abandoned, out on a service call which he might call in to verify if he can help, and once he figures out you are a civilian he will wake you up. The best case scenario is you are woken up at 2am and told to find someplace else to camp. It only takes a phone call to run license plates.
As for lights, the only red lights you can have on the vehicle must be on the rear of the vehicle. Any red lights on the side or front of the vehicle are a violation. I painted mine purple (goes with our Washington Huskies theme).
As for getting identified as an ambulance, it'll happen by the general public. We have Washington Husky decals all over it, a large purple stripe on the body, and a large football helmet mounted on the cab where the light bar was. We were still stopped when we were in traffic from a frantic family looking for their family member who had been run over on his bicycle in the area. They thought we were responding even with our dog sitting shotgun.