Heat 1 teaspoon of the oil in a wide nonstick frying pan or wok over medium-high heat. When oil is hot, add half each of the chopped onion, mushrooms, squash and thyme. Stir fry until squash is hot and bright in color (about 3 minutes). Add 1/4 cup water and 1 1/2 tablespoons of the lemon juice to pan; cover and cook until vegetables are just tender to bite (about 3 minutes). Uncover and continue to cook, stirring until liquid has evaporated (about 3 minutes more). Remove vegetables from pan and set aside.
Repeat to cook remaining chopped onion, mushrooms, squash and thyme, using remaining 1 teaspoon oil; add 1/4 cup water and remaining 1 1/2 tablespoons lemon juice after the first 3 minutes of cooking.
Return all cooked vegetables to pan; gently stir in tomatoes. Transfer vegetables to a serving dish; sprinkle with green onions, cheese and olives.