In the scene where the medic gets shot, watch the shot where Upham brings the bags up to the injured medic. The next shot there is some fog which then reveals the injured medic. In this shot, watch carefully as the cast rips away the medic's shirt. If you look near the neck, you can see the fake stomach vest he is wearing for a split second - when the actor realizes he ripped too far up, he quickly covers it back up.
The story James Ryan tells Miller, about his brothers in the barn with Alice Jardin, was not in the script. Matt Damon ad-libbed it and Spielberg decided to leave it in the final film.