The original 'bog' is Chicken Bog, and its origins are in early South Carolina.
Rice was originally planted in the Carolinas in late 17th century (from East African [probably Madagascar] rice brought over by a sea captain), and 'Carolina long-grain rice' became the most popular in the U.S. South Carolina was the major U.S. rice grower until the Civil War. The 'low country' where rice was grown in the Carolinas were bogs (rice grows best when half submerged in water). So they were familiar with the term 'bog'.
Chicken Bog got its name because the chicken is 'bogged' down in the rice, or because the whole thing was a 'boggy, soggy mess.'