Preheat the oven to 375 degrees F (190 degrees C). Generously grease a 12-cup muffin tin with cooking spray.
Combine egg whites, Greek yogurt, salt, black pepper, garlic powder, onion powder, oregano, and cayenne in a bowl; whisk until smooth and no lumps remain. Mix in red bell pepper, spinach, and Cheddar cheese.
Divide egg mixture between the 12 muffin cups, filling until almost full.
Bake in the preheated oven until egg bites puff up substantially and begin to brown, 20 to 25 minutes. Allow to cool for 5 minutes; bites will deflate substantially at this time. Run a knife around the edges and bottoms to loosen and remove from the cups. Allow to cool to room temperature.