Heat the sesame oil in a large pot over medium heat. Add the scallion whites, ginger and garlic; cover and cook, stirring occasionally, until softened, about 3 minutes.
Add 5 cups water, the chicken broth, chicken breasts, teriyaki sauce and 1/2 teaspoon salt. Increase the heat to medium high; cover and bring to a simmer. Add the squash and reduce the heat to medium low. Cook, covered, until the chicken is just cooked through, about 7 minutes. Remove the chicken to a plate; let cool slightly, then shred.
Return the soup to a simmer over medium-high heat. Add the noodles and bell pepper; cook until the noodles are just tender, about 3 minutes. Add the spinach and shredded chicken. Top with the scallion greens.