Heat vegetable oil in a large stock pot over medium heat. Brown the pork cubes in the hot oil, about 10 minutes. Cook and stir onion and garlic with pork until onion is pork is no longer pink in middle, about 5 minutes.
Bring 2 cups chicken broth and 2 cups beef broth to a boil in a separate saucepan. Stir rosemary, sage, bay leaves, and thyme into broth mixture; continue cooking at a boil until broth is reduced by half, about 10 minutes. Pour reduced broth into pork mixture along with remaining 2 cups chicken and beef broth. Bring to a boil.
Stir butternut squash, potatoes, carrots, and celery into soup. Reduce heat to low and simmer until squash is tender, 10 to 15 minutes.
Stir apples into soup and simmer until tender, about 10 minutes; season with salt and black pepper. Remove pot from heat and let soup cool for 10 minutes before serving.