Preheat an oven to 375 degrees F (190 degrees C). Grease 24 mini muffin cups or 12 standard muffin cups.
In a mixing bowl, cream together the butter and brown sugar until smooth. Mix in the mashed bananas, squash, carrots, and eggs. Stir in the flour, oat bran, baking soda, pumpkin pie spice, and salt until just combined. Spoon the batter equally into the prepared muffin cups.
Bake in the preheated oven until a toothpick inserted into the center comes out clean, 15 to 20 minutes. Cool in the pans for 10 minutes before removing to cool completely on a wire rack. Store at room temperature for up to two days, or freeze.