Place pork (in 1 piece), chicken broth and vinegar in a slow cooker. Cover and cook for 8 hours on low.
Move pork to a plate and, using 2 forks, shred meat and transfer it to a large saucepan. Strain liquid; add 1 cup of it to pork. (Reserve remaining juices for later use.) Stir barbecue sauce into pork.
Place cabbage, yogurt, mayonnaise, vinegar and sugar in a large bowl; toss well.
Reheat pork, adding juice for desired moisture. Serve with coleslaw.