1. Trim the ends off the squash and dice into half-inch cubes. Combine with one tablespoon salt in a colander and let drain in the sink for 10-30 minutes.
2. Add oil to skillet and heat to medium heat. Add the onions and paprika and cook, stirring, until just tender.
3. Squeeze as much liquid as possible from the squash with your hands and add it to the skillet along with the tomatoes (and tomato sauce, if using). Cook, stirring occasionally, until the mixture is tender and no longer releasing liquid, about 10 minutes. Remove the skillet from heat and stir in feta, reserving one tablespoon, and basil.
4. Smooth the mixture evenly in the skillet and make six to eight wells using the back of a spoon. One at a time, crack an egg into a small bowl and pour into one of the wells. Sprinkle salt, pepper, and paprika over each egg.
5. Bake until egg whites are set and yolks are still soft, about six to 10 minutes. Garnish with basil and reserved feta.