For the greens: Place a large saute pan over medium-high heat. Add the bacon and cook, stirring frequently, until crispy and browned, 8 to 10 minutes. Set the bacon aside on a paper towel-lined plate. Add the kale to the pan and toss in the bacon fat. Add the vinegar, 2 tablespoons of water and a pinch of black pepper. Cook, stirring frequently, until wilted down, about 5 minutes. Remove from the heat and stir in the bacon. Set aside for sandwich assembly.
For the squash: Line a baking sheet with parchment paper and spray with olive oil spray. Spread out the slices of squash on the baking sheet and spray the squash generously with the olive oil spray. Sprinkle with the garlic salt, black pepper, paprika and cinnamon. Roast until softened, 45 to 55 minutes. Set aside for sandwich assembly.
For the sandwiches: Lay 2 slices of bread on a work surface. Place a slice of American cheese on a bread slice. Spread 1/4 cup of the greens over the cheese. Place 2 pieces of squash on top. Cover with a piece of pepper jack cheese. Top with a slice of bread. Repeat to assemble 3 more sandwiches.
Place a large nonstick pan over medium heat and melt the butter. Put the sandwiches in the pan and press down with a spatula. Cook until the cheese begins to melt and the bread turns golden brown, about 2 minutes per side. Slice in half.