Make the dry rub: In a small bowl, whisk together all the dry rub ingredients and set aside.
Make the pulled pork: In a medium pot, combine the apple juice, vinegar, sugar and salt over medium heat. Whisk until the sugar and salt have dissolved, then remove from the heat and let cool completely. Completely submerge the pork in the pot, cover in plastic wrap and refrigerate for 1 hour. Preheat the oven to 300º.
Remove the pork from the brine and pat dry with paper towels. Place on a baking sheet and rub the entire surface with the dry rub. Bake until the pork is tender when pierced with a fork, 2½ to 3 hours. Let cool completely, then place into a large bowl. Using 2 forks, shred the pork into thin strands. Stir in the barbecue sauce until incorporated.
Make the egg rolls: Lay out the egg roll wrappers on a clean surface. Spoon 3 tablespoons of pulled pork along the center of each wrapper horizontally and top each with 1 tablespoon of coleslaw. Brush the edges with water and fold the two opposite sides perpendicular to the filling toward the center. Roll each into an egg roll and arrange on a plate, seam-side down.
In a 6-quart Dutch oven, heat 2 inches of vegetable oil to 375º and line a baking sheet with paper towels. Working in batches, fry the egg rolls until golden brown, 3 to 4 minutes. Transfer the egg rolls to the prepared baking sheet to drain.
Transfer to a platter and serve with peanut sauce.