The other side says: "W. Johnston * Die Sinker - Cin, O." Other tokens are cataloged with this die/maker, but not in combination with the "Union" side. The ones I have seen have a Masonic symbol. I have written to Russell Rulau, author of The Standard Catalog of United States Tokens 1700-1900 to see if he can shed any light on this variety.
My searches in the 1860, 1870 and 1880 Census records have not turned up our die maker Johnston. Maybe his name can be found in some Cincinnati City Directories of the era. Another token is seen with an Indian head design dated 1863. The advertising side is a different design and gives an address of 154 Everett St.
Update: William and John S. Johnston are listed in the 1862 Cincinnati City Directory as die sinkers at 429 Central Avenue. They also live at that address. William Johnson is listed in the 1863 Directory at 154 Everett Street as a machinist.