Combine vinegar, mustard, shallot, 3/4 teaspoon salt, and 1/2 teaspoon pepper in a small bowl. Pour in oil in a slow, steady stream, whisking constantly.
Using a mandoline or vegetable peeler, shave squash and carrots into a large bowl. Add dates, sunflower seeds, arugula, and vinaigrette. Toss thoroughly to coat, and season with salt and pepper. Serve immediately.