Combine all marinade ingredients in large plastic resealable food bag; add chicken pieces. Tightly seal bag. Turn bag to coat chicken well. Place into 13x9-inch pan. Refrigerate, turning occasionally, at least 3 hours. Drain; reserve marinade.
Heat one-side of gas grill on medium or charcoal grill until coals are ash white. Place coals in charcoal grill to one-side. Make aluminum foil drip pan; place opposite heat.
To assemble kabobs, alternately thread chicken, mushrooms, tomatoes, bell pepper and squash onto skewers. Brush kabobs with marinade. Place kabobs on grill over drip pan. Grill, turning and basting occasionally, until chicken is fork tender (15 to 20 minutes).