Preheat the oven to 350°F (175°C). Butter a 9- or 10-inch round cake pan.
In a medium bowl, combine the flour, baking powder, baking soda, salt, and cinnamon.
In a large bowl, whisk together the brown sugar, oil, eggs, and vanilla. Stir in the squash and bananas. Add the dry ingredients and stir until just combined.
Pour the batter into the prepared pan and bake until a toothpick inserted in the center comes out clean, 45 to 60 minutes. Turn the cake out onto a wire rack and let it cool completely, about 3 hours.
Make the cream cheese frosting
In the bowl of an electric mixer fitted with the paddle attachment, beat the cream cheese and butter until fluffy and smooth, 3 to 5 minutes. Add the vanilla and confectioners' sugar and beat to combine.
Assemble the cake
Spread the frosting on top of the cake and, if desired, also coat the sides. Otherwise, lick the frosting from the bowl in the kitchen later, when you're cleaning up the empty plates and forks after your guests demolish this cake.