For the meringue: Preheat the oven to 150 degrees F. In a stand mixer fitted with the whisk attachment, whisk the egg whites until frothy. Add the cream of tarter and continue to whisk until soft peaks form. Once the whites are at soft peaks, slowly add in the granulated sugar and continue whisking until stiff peaks form and the egg whites are smooth and glossy.
Place a nonstick baking pad on an upside-down baking sheet and then spray with nonstick cooking spray. Scoop the meringue into a square shape using a 3-inch square stencil. Smooth out the edges and make the squares even with a long, flat icing spatula. Make as many squares that can fit onto the baking sheet. You'll need 6 sides to make a full box, so make sides based on how many boxes you want to make. Remove the stencil. Bake the meringue until set and not tacky to the touch, 3 hours. The meringue should have no color on it after baking.
For the diplomat cream: In a saucepan, bring the milk to a light simmer. In a bowl, lightly whisk the squash puree, 1/4 cup of the granulated sugar, the cornstarch, flour, cinnamon, cloves, nutmeg and yolks until smooth. Add half of the steaming milk into squash mixture very slowly, whisking as you pour. Whisk in the remaining milk. Return the mixture to the saucepan and cook over medium heat, whisking constantly, until thick, 3 to 4 minutes. Remove from the heat and add the cubed butter and vanilla, mixing until the butter melts. Cover the mixture with plastic wrap, lightly pressing the plastic against the surface to prevent a skin from forming. Cool the pastry cream completely.
Once the pastry cream is completely cool, place the heavy cream and the remaining 1 tablespoon granulated sugar in a stand mixer fitted with the whisk, and whip until stiff. Fold the whipped cream into the cooled pastry cream to create the diplomat cream. Keep refrigerated.
Use royal icing as the glue between the sides of the box, leaving one side open. When the box is set, spoon the diplomat cream into the box and top with the final side as the lid. Dust with cinnamon and confectioners' sugar to garnish, and then serve.
If you'd like to decorate the meringue sides, do so before building the boxes.
Please note: these recipes have not been tested by Food Network, so results may vary.