It's implied in the shows that he has a backstage romance with fellow Country Bear performer Teddi Barra.
Behind the Scenes
In the early Bear Band treatments of the attraction, Henry was only known as the "MC Bear". Henry would derive his name from Henry Jaynes, half of a hillbilly musical comedy duo called "Homer and Jethro", whose songs would be used for Henry and Wendell's segments of the show. By taking the name Henry, this allowed the Imagineers to keep much of the written banter of the original songs intact.
From 1972's Country Bear Jamboree record album:
Henry, master of ceremonies at the Country Bear Jamboree, stands six feet tall in his stocking paws. He is another famous football player who entered show business. Henry was formerly with the Goose Creek Bruins. One day they tried a hidden ball play, and Henry hid the ball so well he couldn’t find it. This hastened his transfer to music. For a while, he had trouble finding a melody, too. But Henry is a likable sort, and audiences go for him like he goes for honey. For a finale, Henry sings Ballad of Davy Crockett with Sammy, his live raccoon hat. Henry is glad he turned in his football gear for a starched shirt and Sammy. “After all,” he explains, “I never knew a football helmet that could sing like Sammy.”