Cook pasta according to package directions, omitting salt and fat. Add asparagus and peas during the last minute of cooking. Drain; rinse under cold running water. Drain well; place in a large bowl. Add 1 tablespoon oil; toss to coat. Stir in red cherry tomatoes, yellow cherry tomatoes, red bell pepper, yellow bell pepper, orange bell pepper, squash, and chives.
Combine remaining 3 tablespoons olive oil, juice, salt, black pepper, and garlic in a small bowl; stir with a whisk. Pour dressing over pasta mixture, and toss well. Serve at room temperature or chilled.