Preheat oven to 350°F. Peel and cut squash lengthwise into two halves. Clean squash and keep seeds; rinse seeds and pat dry. Toss together brown sugar, salt, cayenne, and 3 tablespoons butter. Add seeds and toss to coat. Toast seeds in 1 layer in a nonstick shallow baking pan in middle of oven until golden, about 10 minutes, then cool. Toss squash with remaining tablespoon butter mixture and salt and pepper to taste in baking pan and roast squash until just tender, about 20 minutes. Cool for a few minutes and then cut squash into thin 1/2 inch slices.
Whisk together oil, lemon juice, and salt and pepper to taste in a large bowl. Crumble cheese into bowl, then add arugula. Gently toss in squash and sprinkle with toasted seeds. Serves 4.