Pat the hash browns dry and add to a large bowl. Melt 3 tablespoons of the butter in a medium skillet and add to the hash browns with salt and pepper and mix well. Press the hash browns on the bottom and up the sides of a 9-inch pie pan, pushing them gently into the corners. Bake until golden brown, 20 to 25 minutes. Remove the crust from the oven and reduce the oven temperature to 375 degrees F.
Meanwhile, melt the remaining tablespoon butter in the same skillet over medium-high heat. Add the zucchini and bell pepper and cook, stirring, occasionally, until slightly softened, about 3 minutes.
Spoon the vegetables into the crust and sprinkle with the ham and Cheddar.
Whisk the cream, chives, cayenne and eggs together in a bowl with 1/4 teaspoon salt. Pour the mixture into the crust. Bake the quiche until golden brown and just set in the center, 20 to 25 minutes.
Cool at least 10 minutes before serving. Serve warm or at room temperature.