Put chicken, onion, and garlic in a large stockpot with enough water to cover; bring to a boil. Cook until the chicken is no longer pink at the bone and the juices run clear, about 1 hour. An instant-read thermometer inserted into the thickest part of the thigh, near the bone should read 165 degrees F (74 degrees C). Drain and reserve liquid. Set chicken aside to cool. Remove and discard bones from chicken. Chop the chicken meat.
Put tomatoes in a large stockpot with enough water to just cover; add carrots, potatoes, zucchinis, yellow squash, green bell pepper, yellow bell pepper, red bell pepper, and chayote squash. Bring the water to a boil, reduce heat to medium-low, and cook until the vegetables are tender, about 30 minutes.
Pour reserved chicken stock into the pot. Stir chopped chicken, cilantro, and tomato bouillon into the liquid. Bring the mixture to a simmer and cook another 15 to 20 minutes.