A box body is normally an enclosed body with a roof. At its simplest, it's a van. Most of those fitted to military soft skins house radio equipment, workshops, printing presses and other things that need protection from the weather or better insulation than a canvas topped lorry could offer. A common British military term for this type of body is "house type".
Open bodies like pick-ups are often called cargo or GS bodies but there are no fixed names for vehicle bodies. British and American expressions often differ so that, for example, an open topped pick-up body might be called a cargo box in US parlance.