For the pork: Combine the coffee, brown sugar, chipotle powder, salt, black pepper, cumin and garlic powder in a medium blow. Rub it all over the pork. Refrigerate for at least 1 1/2 hours and up to overnight. Remove from the fridge and bring to room temp for 1/2 hour.
Preheat the oven to 350 degrees F. Scatter the onions and bay leaf in the bottom of a Dutch oven. Put the pork on top of the onions. Pour in the rum and 3/4 cup water. Cover and braise in the oven. Check the liquid level at 2 1/2 hours, add 1/4 cup more water if needed. Continue to cook until the internal temperature is 190 degrees F, about another 30 minutes. The pork should cook about 3 hours total. Remove from the oven and let cool. Shred the meat, discarding any skin or fat. Set aside to build the sandwiches.
For the beer cheese squash soup: Heat a heavy-bottomed soup pot over medium-high heat. Add the oil, onion, garlic and 1/8 teaspoon salt, and cook, stirring, until soft but no color, 4 to 5 minutes. Add the squash and chicken stock, bring to boil, and then reduce to a simmer. Cook until very soft, 8 to 10 minutes. Add the red pepper flakes. Let cool slightly, then carefully puree the soup in batches in a blender. Alternatively, use an immersion blender to puree. Return the soup to the pot, add the half-and-half, beer, nutmeg and the remaining salt. Stir in the Cheddar until melted.
For the pickled carrots: Heat the vinegar with the garlic and 1/4 teaspoon salt in a small saucepan until it comes to a boil. Remove from the heat and add the carrots. Let sit at room temperature 30 minutes. Remove the carrots and reserve. Save 1/4 cup plus 2 tablespoons of the vinegar mixture for the chimichurri.
For the chimichurri: Combine the parsley, oregano, reserved vinegar mixture, crushed red pepper and remaining 1/2 teaspoon salt in the bowl of a food processor. Add the oil and pulse to combine.
For the Cubano croutons: Heat a flat griddle over medium heat. Sear the Canadian bacon on both sides until slightly caramelized. Remove to a plate and set aside. Slice a roll in half. Add 1 piece of Gruyere, 2 slices bacon, 1/3 cup shredded pork, a thin layer of the carrots, 1 tablespoon of the chimichurri and another slice of cheese. Top with the top of the roll. Repeat with the remaining rolls.
Put the sandwiches on the griddle over medium heat. Press the sandwiches with a heated cast-iron pan on top to flatten, and cook until crisp, brown and the cheese melts, 4 to 6 minutes. Remove from the heat to a cutting board.
To serve: Ladle 8 ounces of the beer cheese squash soup into 4 shallow bowls. Cut the sandwiches into 1-inch croutons and float them in the soup. Serve with the crushed plantain chips sprinkled on top. Reserve the remaining pork, soup and carrots for another use.
When blending hot liquid, first let it cool for five minutes or so, then transfer it to a blender, filling only halfway. Put the lid on, leaving one corner open. Cover the lid with a kitchen towel to catch splatters, and pulse until smooth.