In a small container, mix all the spice mix ingredients and set aside.
Cut chicken into 1-inch bite-sized pieces and place in a glass baking dish.
Slice squash into 1/4-inch discs.
Cut chayote into 1/4-inch discs, then into quarters, and place all vegetables in another dish.
Sprinkle half the spice mix onto the chicken, and the other half onto the vegetables.
Divide lime juice between both bowls.
Add 1/2 cup olive oil to the chicken and 1/2 cup to the vegetables. Reserve the remaining 1/2 cup.
Mix each bowl well, coating all pieces thoroughly. Add more oil if needed (up to 1/4 cup more for chicken).
Cover both dishes and refrigerate for at least 6 hours, or overnight.
Soak wooden skewers in water for at least 30 minutes before grilling, then drain.
Cut bell pepper into 1-inch chunks.
Assemble skewers: Start with squash, then 2 pieces of chicken, then chayote, then 2 more chicken pieces, bell pepper, and repeat.
Spray grill with cooking spray. Preheat grill to 350°F (medium heat), then reduce to 250°F.
Sprinkle assembled skewers with salt and black pepper.
Grill skewers, turning occasionally, until chicken is fully cooked and reaches an internal temperature of 165°F.
Transfer to a serving platter and enjoy!
