Bring chicken broth, chicken breasts, onion, carrots, turnip, and celery to a boil in a large pot until chicken is no longer pink in the center, about 20 minutes. Season with salt and pepper. Remove chicken from broth; strain and discard vegetables. Add green beans and rice; continue to simmer until rice is tender, about 20 minutes more.
Cut chicken into chunks. Stir chicken and cream cheese into broth until cheese is melted, about 5 minutes.