Put squash in a saucepan and pour in enough water to cover; bring to a boil and cook until tender, about 15 minutes. Drain water from saucepan, cool squash, and mash using a potato masher or fork.
Pour warm water into a bowl and sprinkle in yeast. Let stand until the yeast softens and begins to form a creamy foam, about 5 minutes.
Combine flour, white sugar, and salt in a large bowl. Stir yeast mixture, mashed squash, milk, and 1/2 cup butter into flour mixture until dough starts to hold together. Turn dough onto a lightly floured work surface and knead until smooth.
Place dough in a lightly oiled bowl, turning to coat all sides. Cover bowl with a damp cloth and let dough rise in a warm place until doubled in size, about 1 hour.
Preheat oven to 400 degrees F (200 degrees C). Lightly grease a 9x13-inch baking dish.
Roll dough into a 16x21-inch rectangle. Spread 1/3 cup butter over the top of dough. Mix brown sugar and cinnamon together in a bowl and sprinkle over butter layer. Roll dough around filling and cut into 12 to 15 rolls. Place rolls in the prepared baking dish and let rise until nearly doubled in size, about 30 minutes.
Bake in the preheated oven until golden brown, about 15 minutes.
Beat confectioners' sugar, 1/4 cup butter, cream cheese, and vanilla extract together in a bowl using an electric mixer until frosting is smooth; spread onto warm rolls.
