Johnson is a populated place situated in Cochise County, Arizona, United States, on the east side of the Little Dragoon Mountains. The locatio…Johnson is a populated place situated in Cochise County, Arizona, United States, on the east side of the Little Dragoon Mountains. The location began as a mining camp, and a post office was established there in 1900, with William De H. Washington as its postmaster. The post office closed in 1929.