Preheat oven to 400°F. Halve squash, scoop seeds, roast cut-side down 35–40 minutes until tender; shred with fork.
Season chicken with salt and pepper, dredge in flour, shaking off excess.
Heat oil in a skillet over medium-high heat; cook chicken 3–4 minutes per side until golden and cooked through. Transfer to a plate.
Reduce heat, add butter, broth, lemon juice, and capers; simmer 2 minutes, scraping browned bits.
Return chicken to pan to warm and coat with sauce; sprinkle parsley.
Serve chicken slices over spaghetti squash and spoon extra sauce on top.