Preheat the oven to 400 degrees F.
Place zucchini, squash, corn, and tomatoes on a large-rimmed sheet pan. Drizzle with olive oil, and season with garlic powder, salt, black pepper, and lime juice.
Transfer to the oven and roast for 15 minutes.
Combine the chicken, panko, red onion, egg, ¼ Cup BBQ sauce, salt, chili powder, garlic powder, and pepper in a large mixing bowl.
Using a cookie scoop, roll the mixture into 1½ or 2 Tablespoon balls.
Remove the sheet pan from the oven and use a spatula to push the vegetables to one side, or spread out across the entire sheet pan. Set the oven temperature to 425°F.
Place meatballs on the other side of the sheet pan, or on top of the vegetables.
Bake for 10 to 15 minutes. Baste the meatballs with the remaining BBQ sauce every 5 minutes.
Garnish with crumbled feta and chopped basil and serve.