Bring mirin, low sodium soy sauce, honey and crushed garlic to a boil in a medium-sized sauce pan, and cook over medium heat for about 5 minutes. Set aside to cool.
Cut the chicken into 1-inch pieces and place in a ziplock bag; pour half of the marinade over the chicken. Place the zucchini in a second large ziplock bag and pour the remaining marinade over the zucchini.
Refrigerate for at least 30 minutes. Meanwhile, soak the skewers in water 30 minutes so they don’t burn.
Thread the chicken onto skewers, alternating with green onion so that each stick has 3 cubes of chicken and two pieced of green onion, discarding the chicken marinade.
Thread the zucchini onto skewers, alternating with remaining green onion, reserving the marinade for basting.
Preheat the grill or a grill pan over medium-high heat. When hot, spray with oil then reduced heat to medium; grill the zucchini and chicken skewers about 5 to 6 minutes on each side brushing both sides of the skewers with the yakitori sauce during the last few minutes of cooking time.