Combine butternut squash, potato, onion, and salt in a large bowl; let sit until moisture is drawn out, about 10 minutes. Transfer to a colander and squeeze out excess moisture.
Place squash mixture in a bowl and mix in flour, egg, black pepper, and baking powder until well blended.
Heat 1 tablespoon olive oil in a skillet over medium heat; cook and stir spinach until wilted, about 2 minutes. Season spinach with salt and pepper; transfer to a bowl.
Scoop about 1 1/2 tablespoons squash mixture and form into a small ball using your hands; flatten into a patty. Add 1 teaspoon spinach to the flattened patty and form patty around the spinach into a ball. Flatten the ball with the palms of your hands. Repeat using the remaining squash mixture and spinach creating 12 to 14 more spinach-stuffed patties.
Heat 1 tablespoon olive oil in a large skillet over medium heat; cook patties until lightly browned, 4 to 5 minutes per side. Transfer cooked patties to a plate and repeat with remaining oil and patties.