Drain artichoke hearts, reserving liquid in a large bowl.
Bring mirlitons and water to cover to a boil in a Dutch oven, and boil 45 to 50 minutes or until tender. Drain and cool. Peel mirlitons. Cut in half lengthwise; remove seeds, if desired. Cut mirliton halves into 1/4-inch-thick slices.
Whisk together reserved liquid, basil, and next 5 ingredients. Add mirlitons, artichoke hearts, and bell pepper; toss well. Cover and chill 2 hours, stirring occasionally. Serve over salad greens.
* 6 large yellow squash or zucchini may be sustituted. Cut in half lengthwise; cut halves crosswise into 1/4-inch-thick slices. Cook in boiling water to cover 5 minutes or until tender; drain, and cool.