Preheat oven to 375°. Cut tops (stem ends) off squash and reserve. Scoop out and discard insides of squash, leaving a shell at least 1/4 in. thick. Brush squash and tops with about 1 tbsp. olive oil, set on a baking sheet, and bake until tender when pierced with a fork, 15 minutes.
Meanwhile, heat remaining 1 tbsp. olive oil in a large frying pan over medium-high heat. Add onion and 3/4 tsp. salt. Cook, stirring occasionally, until onion is soft, 3 to 5 minutes. Add garlic and cook, stirring, until fragrant, 30 seconds. Add thyme and cook, stirring, until combined.
Set squash tops aside. Divide onion mixture evenly among the squash. Crack 1 egg into each squash, lifting out any white that doesn't fit. Sprinkle eggs with remaining salt. Bake until eggs are set, 15 to 20 minutes. Serve immediately, with tops replaced or set to the side.
Note: Nutritional analysis is per squash.