For the pork: Preheat the oven to 350 degrees F.
Make small holes all over the pork skin with a thin sharp knife; this will help the skin to render during cooking. Generously sprinkle pork all over with salt and pepper. Add the pork, apple cider, white vinegar and garlic to a Dutch oven or heavy-duty pot. Make sure to massage the liquid all over the pork. Cover with a tight fitting lid and place in the oven, skin-side up, for 3 hours total cooking time. After 1 1/2 hours, turn the pork over, skin-side down, and baste with the cooking juices. Continue to cook, covered, for the remaining 1 1/2 hours. The pork should be fork tender and golden brown around the edges. The cooking liquid will have reduced and become richer in color. Let the pork rest, uncovered, in the pot.
For the BBQ sauce: Heat the vinegar and sugar in a medium saucepan over medium heat until the sugar dissolves. Off the heat, stir in the ketchup, honey, red pepper, salt and black pepper.
When the pork is cooled, shred the meat, including the skin, by hand or with forks. Run a knife though the meat once or twice until the strands are about 1/2-inch long. Recombine the chopped meat with the braising juices and garlic. Add a desired amount of the BBQ sauce, depending on how much vinegar you want (you will have 2 cups of sauce) and serve the rest on the side. Spoon the pulled pork on a bun and top with slaw.