VERSION #1: Trim the ends off the squash and grate using a box grater or a food processor with the grating attachment. Combine with 1 tablespoon salt in a colander and let drain in the sink for 30 minutes. Squeeze as much liquid as possible from the grated squash. Add the squash to the skillet along with the tomatoes. Cook, stirring occasionally, until the mixture is tender and no longer releasing liquid, about 10 minutes. Proceed with recipe in Step #3.
VERSION #2 (my change): Trim the ends off the squash. Dice the [unpeeled] summer squash into 1/2" cubes. Cut the zucchini and crookneck into 1" slices. In a skillet, heat 2 tablespoons olive oil over medium heat. Add the diced summer squash to the pan and saute until lightly browned, about 5 minutes. Next add the shallots, garlic, and paprika and cook, stirring, until just tender and fragrant. Reduce heat to medium-low then stir in the tomatoes. Cook, stirring occasionally, until the mixture is tender and no longer releasing liquid, about 8-10 minutes. If mixture starts to dry out too much, reduce heat. Proceed with recipe in Step #3.
Remove skillet from heat and stir in 1/4 cup of the basil.
Smooth the mixture evenly in the skillet and make four wells using the back of a spoon. Pour 1/2 teaspoon olive oil in each well. 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. Arrange small bits of the chevre around and in between the eggs.
Cover and cook over low-medium heat until egg whites are set and yolks are still soft, about 10 minutes. I checked the eggs after 5 minutes, then cooked them another 2 minutes for our preference.
Garnish eggs with basil and serve with crusty bread. A simple green salad and lightly seasoned black beans made the meal complete. I threw together a quick arugula salad. You could also serve the squash and eggs over raw greens as a suggestion.