Build a fire (wood or a combination of charcoal and wood) for indirect cooking by situating the coals on only one side of the grill, leaving the other side void.
In a small bowl, combine the rub ingredients. Mix well and set aside.
In a separate mixing bowl, combine the juice, water, sugar, salt, and Worcestershire; blend until the sugar dissolves. Using a meat syringe, inject the meat evenly every 1-inch, using the entire amount of the injection solution.
Vinegar Sop Mop
Combine all ingredients in a bowl and set aside.
Smoking the Pork Butt
Apply the dry rub to the meat in an even coating, patting so the rub adheres.
When the heat reaches 225 degrees F (107 degrees C), place the pork shoulder, skin side down, on the void side of the grill, and close the lid. Cook for 14 to 16 hours, adding charcoal as needed during the cooking process to keep the cooker temperature at 225 degrees F. Two small wood chunks should be added every hour to increase the smoke flavor.
When the meat has cooked for 13 hours, start basting the shoulder with the Vinegar Sop Mop every hour. When done, the blade bone (the bone visible on the side of the shoulder) should release with a firm tug and the internal temperature of the meat should reach 195 degrees F (91 degrees C).
Remove the pork to a cutting board and let stand for 30 minutes. Wearing insulated rubber gloves, pull the pork from the bones by hand. Pull off and discard all visible fat.