Preheat oven to 350 degrees F (175 degrees C). Grease an 8-inch square baking dish.
Heat 1 teaspoon vegetable oil in a skillet over medium heat; cook and stir onion, bell pepper, and garlic in hot oil until just tender, about 5 minutes. Transfer mixture to a large bowl.
Mix eggs, zucchini, yellow squash, carrot, green onion, baking mix, 1/2 cup Cheddar cheese, 1/4 cup vegetable oil, salt, Italian seasoning, paprika, red pepper flakes, and black pepper into onion mixture. Spread vegetable mixture into prepared baking dish and sprinkle with remaining 1/2 cup Cheddar cheese.
Bake in preheated oven until lightly browned, about 35 minutes.