Whisk together the water, salt, sugar, and peppercorns in an extra large stockpot or large plastic container until most of the salt and sugar have dissolved. Place the pork shoulder in the mixture, making sure it is completely submerged, and refrigerate for at least 12 hours and up to 24 hours. Remove pork, rinse well with cold water then pat dry. Season pork with cilantro and achiote seasoning.
Prepare a Caja China according to the manufacturer's directions. Place pork, skin side down, in the box and carefully fill tray to its full capacity with lit charcoal. Cook for 1 hour, carefully uncover (you will need to protect yourself from the heat) the Caja China, and turn the meat over. When you turn the pork shoulder, make some small cuts in the skin. Cover the Caja China, add more charcoal as needed, and continue roasting for 2 to 2 1/2 more hours. Do not uncover until pork is done. Alternatively, you can grill the pork; start out on high heat to char it and then reduce the heat, cover, and cook until desired degree of doneness. You can also use a rotisserie.
When the pork is done, remove it from the Caja China and let rest for at least 15 minutes before slicing. Slice and serve with the Sour Orange-Habanero Dipping Sauce.
Sour Orange-Habanero Dipping Sauce:
Heat grill to high. Bring orange juice and lime juice to a boil in a medium non-reactive saucepan on the grates of the grill and cook until reduced to about 2 cups. Let cool slightly. Transfer the mixture to a blender and add the onion, garlic, habanero, vinegar, and honey and blend until smooth. With the motor running, slowly add the oil and blend until emulsified. Season with salt and pepper, to taste. Let cool to room temperature before serving.