Preheat grill, or grill pan to medium-high heat. Cut squash lengthwise into 1/4-inch-thick slices. Spray pan with cooking spray and grill 2 minutes each side. Remove and cut into 1-inch pieces. Cut roasted bell peppers into squares and add to bowl.
Pound chicken to an even thickness. Combine 2 tsp oil and 1 tbsp vinegar and brush onto chicken. Season with salt and pepper. Add to the grill until cooked through (165 F internal temp). Cover to keep warm.
Whisk remaining 1 tbsp oil with 2 tbsp vinegar, mustard, salt, and pepper in a small bowl. Pour over squash and peppers. Add shallots and toss (optional but delicious). Add arugula and spinach mixture. Toss the salad together and top with chicken.