How certain are you of the spelling of "Brinkey"? And the rank? Here are two possibilities:
Charles Brunke, 24th Mich. Inf. (Co.B). Born ca. 1844; rose to the rank of corporal.
William Brunke, 19th Mich. Inf. (Co.I). Born ca. 1839; remained a private.
Both of thses men lived in Royalton, Berrien County, Michigan. In the 1860 census, the spelling of their last name is rendered "Brinker."