In a large saucepan, cover peeled and chopped squash with water. Bring water to a boil and cook until tender, about 15 to 20 minutes. Let cool and mash. Reserve 1 cup for use in this recipe and freeze remainder for later use.
In a small bowl, dissolve yeast in warm water. Let stand until creamy, about 10 minutes.
In a large bowl, combine the yeast mixture with milk, butter, 1 cup mashed squash, 1 egg, brown sugar, salt and 2 cups flour; stir well to combine. Stir in the remaining flour, 1/2 cup at a time, beating well after each addition. When the dough has pulled together, turn it out onto a lightly floured surface and knead until smooth and elastic, about 8 minutes. Lightly oil a large bowl, place the dough in the bowl and turn to coat with oil. Cover with a damp cloth and let rise in a warm place until doubled in volume, about 1 hour.
Deflate the dough and turn it out onto a lightly floured surface. Divide into 6 equal pieces and roll each into a rope about 18 inches long. Take 3 ropes, pinch ends together and then braid ropes together. Pinch ends to seal. Do the same with the other 3 ropes of dough. Place braids on lightly greased baking sheets. Cover and let rise for about 30 minutes. Meanwhile, preheat oven to 350 degrees F (175 degrees C).
In a small bowl, beat together 1 egg and 1 tablespoons water; brush loaves with egg wash. Bake in preheated oven for 20 to 25 minutes. Remove from baking sheets and let cool on a wire rack.