Add the chicken thighs to a large stockpot and fill with approximately 3-4 quarts of cold water. Bring to a boil. Cover and reduce heat to low and simmer for approximately 30-40 minutes, or until chicken is fully cooked.
As the chicken cooks, chop and prepare the vegetables and then move on to the next step.
Add one tablespoon of olive oil to a large pot or Dutch oven set over medium heat. Add the onions and cook until translucent, approximately 5 minutes. Next, add the carrots and celery and mix well to combine with the onions. Continue to cook for 10-15 minutes, stirring often. Stir in the minced garlic, salt, and black pepper. Reduce heat if necessary to prevent the vegetables and/or garlic from burning.
Check on your chicken and broth. Once the chicken is fully cooked, use kitchen tongs to remove the chicken and set it aside until it is cool enough to shred. Meanwhile, remove the broth from heat. Carefully strain the broth through a fine-mesh strainer into the same pot as the vegetables.
Once the chicken is cool enough to handle, use two forks (or your hands) to shred the chicken.
Add the shredded chicken, lemon juice, bay leaves, zucchini, and yellow squash to the pot. Return to a boil, then reduce heat to low, cover, and simmer for 30 minutes.
Approximately 5-10 minutes before serving, stir in the chopped kale and season with additional salt and pepper, to taste.