Swiss is a town in Burnett County in the U.S. state of Wisconsin. The population was 790 at the 2010 census. The unincorporated community of Danbury is located within the town. Swiss is located in northern Burnett County along the St. Croix River, which forms the Wisconsin–Minnesota border. The Yellow River enters the town from the south and drops 6…Swiss is a town in Burnett County in the U.S. state of Wisconsin. The population was 790 at the 2010 census. The unincorporated community of Danbury is located within the town. Swiss is located in northern Burnett County along the St. Croix River, which forms the Wisconsin–Minnesota border. The Yellow River enters the town from the south and drops 60 feet in just over a mile before joining the St. Croix River at Danbury.