While (mostly) a ''Modern'' Military Clothing term - the ''Bush Hat'' can actually trace it's origins to the mid-19th Century. Most Military Armies still used stiff (and heavy) headgear (both on and off the battlefield). As such with the advent of The French Kepi (soft cap), many Nations began to quickly adopt some version. However, most Kepi's were of wool construction and as such - especialy in the heat, such headgear ''cooked'' the head of a Soldier. Towards the middle of World War II, many Nations that operated in The Pacific Theater of Operations, began to adopt soft canvas headgear - similar to a ''Slouch Hat'', but much better as maintaining coolness and allowing for heat dispersal. These fprms of headgear would be known as ''Bush Hats'', for their sole use in The Jungle (in most cases).
Later, especially in The American Army, and with it's deployment to Southeast Asia in the 1960s, a version of The Bush Hat was developed and became known as a ''Boonie Hat'' (referencing the American slang for The Jungles of Vietnam - eg ''The Boonies''). Very popular with most classes of Military Arms, they would be somewhat retired after The War (1975). Later, Bush Hats would be pressed into ''worldwide service'' by those Nations fighting The Global War on Terrorism, in particular - Desert Warfare. The Bush or Boonie if you prefer, is still regarded as one of The Militaries most comfortable of headgear used in The Field.
Coloring varied according to need (or National Army). Often neutral colors (browns and tans) were popular, followed by variations on the color green. Later, a plethora of ''camo'' (woodland and related) patterns were pressed into usage. Typically, Soliders shunned away from using any form of Rank or Insignia on Bush Hats. The plyable nature of the material used, allowed for a Soldoer to be able to quickly wring out any moisture, which allowed for quick drying. Most such Hats also featured drawstrings which allowed for better securement to the head.