Preheat an oven to 350F and spray a 12-muffin tin with nonstick cooking spray. I used a silicone muffin mold on top of a baking sheet.
Add the drained potatoes and mushrooms to a large skillet over medium-high heat. Sprinkle the salt and peppers over the mixture. Cook until the mushrooms begin to brown slightly and the potatoes begin to crisp around the edges, about 6-8 minutes. (If you'd like crispier potatoes, cook them a few minutes on their own before adding the mushrooms.)
Add the spinach and stir well, spreading out the spinach as evenly as possible. Reduce the heat and cook until everything is evenly mixed and no moisture remains, about 4-5 minutes.
Spoon the mixture into the muffin tin as evenly as possible.
Crumble the feta cheese and sprinkle it over the spinach mixture.
Pour the egg whites over the mixture. If you're using liquid egg whites from a carton, you may want to go straight from the carton for easy pouring. You can place the carton on a food scale and zero out the scale to see how much you're pouring into each slot. (The amount may vary based on the size of your muffin tin.)
Bake for 20-23 minutes or until the egg white cups are fully cooked through. Let the cups cool for a few minutes before removing from the muffin tin. Refrigerate leftover egg cups for 2-3 days.