Heat 1 Tbsp olive oil in a 12-inch non-stick skillet over medium-high heat. Dab chicken dry with paper towels. Season with a fair amount of salt and pepper, 2 tsp Italian seasoning, 1/2 tsp garlic powder, onion powder and lemon zest then toss to coat (I just added the seasonings to the chicken on the cutting board). Melt butter in pan with olive oil then add chicken and cook 3 minutes then flip and cook opposite side until chicken has cooked through (center of larger pieces should register 165 degrees), about 3 minutes longer. Transfer to a plate and tent with foil to keep warm.
Heat remaining 1 Tbsp oil and 1 Tbsp butter in same skillet over medium-high heat. Add zucchini and squash, then season with salt, 1 tsp Italian seasoning and 1/2 tsp garlic powder. Cook tossing occasionally until just tender, about 4 minutes.
Return chicken to pan with zucchini and squash. Drizzle in lemon juice and toss. Sprinkle with parmesan and parsley. Serve warm with more parmesan and lemon zest if desired (I served mine over cooked orzo which is delicious).