Heat oven to 450° F. Cook the noodles according to the package directions; drain and rinse.
Toss the squash, eggplant, and 3 tablespoons of the canola oil on 2 rimmed baking sheets. Roast, tossing once, until tender, 16 to 20 minutes.
Combine the soy sauce, vinegar, sesame oil, brown sugar, and the remaining 2 tablespoons of canola oil in a large bowl. Add the soba, squash, eggplant, and cabbage; toss to combine.
Serve sprinkled with the mint.