For the dry rub: Mix together the paprika, black pepper, salt, brown sugar and cayenne pepper.
For the pork: Sprinkle the dry rub all over the pork tenderloins and place in a resealable plastic bag. Refrigerate for at least 2 hours, and then bring to room temperature before cooking.
Cut each tenderloin into 2 or 3 large pieces. Heat the oil in a large soup pot or Dutch oven. Sear the pork on all sides in batches (you don't want to overcrowd the pan), 4 to 5 minutes. Cover with the apple juice, vinegar, BBQ sauce, garlic and ginger. Bring to a simmer over medium-high heat, and then cover and simmer for 30 minutes, turning occasionally.
Transfer the pork to a clean cutting board and shred using 2 forks. Place the shredded pork back into the pot and continue to cook over medium-low heat, uncovered, just to heat through, 10 minutes longer.
Serve on hamburger buns topped with BBQ sauce and coleslaw with mustard and pickles, if desired.