Preheat oven to 375 degrees F (190 degrees C). Grease 12 muffin cups.
Whisk flour, oat bran, ginger, baking powder, and baking soda together in a bowl.
Place butternut squash into a pot with just enough water to almost cover the squash; bring to a boil. Reduce heat to medium-low and simmer until tender, 10 to 15 minutes. Drain and mash squash with a potato masher until smooth.
Mix squash and apricot jam together in a bowl.
Whisk cream, sugar, and eggs into the squash mixture until light and creamy; stir into flour mixture until just combined. Pour batter into each muffin cup, filling each cup to about 3/4-full.
Bake in the preheated oven until lightly browned, about 20 minutes. Cool muffins in muffin cups before removing.