Heat the oil in a large pot over medium heat; cook and stir the onion and garlic in the hot oil until fragrant. Quickly stir the ginger and fish sauce into the onion and garlic mixture before adding the chicken; cook together for 5 minutes. Pour the chicken broth over the mixture and cook another 5 minutes. Add the chayote to the mixture and simmer until the chicken is no longer pink in the center, about 10 minutes more. Season with salt and pepper. Add the bok choy and spinach; cook until the spinach is just wilted, 1 to 2 minutes. Serve hot.