In a measuring cup, whisk broth, teriyaki, vinegar, cornstarch, sugar and red pepper flakes. Set aside.
Bring a large pot of water to boiling. Heat the canola and the sesame oils in a large nonstick skillet over medium-high heat. Add chicken and cook 4 minutes, stirring occasionally. Remove to a plate. Add mushrooms and cook 2 minutes.
Meanwhile, add soba noodles to boiling water and cook, 3 to 4 minutes. Drain.
Add bok choy and red pepper strips to skillet. Cook 2 minutes. Return chicken to pan along with broth mixture. Cook 2 to 3 minutes, until thickened. Gently stir in soba noodles and serve.