Preheat oven to 375°. Line bottom of a 15x10x1-in. pan with parchment.
In a large bowl, beat eggs on high speed 3 minutes. Gradually add 1 cup sugar, beating until thick and lemon colored, about 2 minutes. Stir in squash. In another bowl, whisk together flour, baking soda and cinnamon; fold into squash mixture. Transfer to prepared pan, spreading evenly. Sprinkle with walnuts.
Bake until a toothpick inserted in center comes out clean, 13-15 minutes. Cool on a wire rack 10 minutes.
Invert onto a tea towel dusted with confectioners' sugar. Gently peel off parchment. Roll up cake in the towel jelly-roll style, starting with a short side. Cool completely on a wire rack.
In a bowl, beat first four filling ingredients until smooth. Unroll cake; spread filling over cake to within 1 in. of edges. Roll up again, without towel; trim ends. Place on a platter; seam side down. Refrigerate, covered, 1 hour. If desired, dust with additional confectioners' sugar before serving.