Preheat oven to 425°F (220°C). In a bowl, combine butter, hash browns, green onions, egg whites, salt and pepper and mix well.
Evenly divide hash brown mixture among a large 6-cup, non-stick muffin tin or six 6-oz (175 g) ramekins. Firmly press mixture into bottoms and up sides of each cup. Bake on bottom rack of oven until golden, about 15 min. Set aside to cool.
In a non-stick skillet, cook bacon over medium-high heat until crispy, about 5 min. Transfer bacon to a paper towel–lined plate and discard all but 1 tbsp (15 mL) of bacon fat. Add red pepper to skillet and cook until softened, about 4 min. Remove from heat and crumble bacon into skillet. Divide red pepper mixture evenly among muffin cups.
Sprinkle egg cups with cheese and then crack 1 egg into each cup. Bake until whites are set but yolks are still runny, about 10 min. (Add 1 to 2 min. for firm yolks.) Season with salt and pepper. Use a knife to remove potato cups from pan and serve immediately.
Make ahead: Prepare steps 1 and 2 the night before, cover and refrigerate. Bring cups to room temperature before proceeding with step 3.
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