Heat grill to medium. In a shallow baking dish or large bowl, toss the squash and scallions with 2 tablespoons of the oil, ½ teaspoon salt, and ¼ teaspoon pepper. Grill the vegetables until cooked through, 4 to 6 minutes per side for the squash and 1 to 2 minutes per side for the scallions. Cut up the scallions; return the vegetables to the dish.
Stretch the dough into four 8-inch rounds. Dividing evenly, brush the dough (on both sides) with 2 tablespoons of the oil. Grill the dough until puffed and cooked through, 2 to 3 minutes per side. Dividing evenly, dollop the ricotta on the pizzas and top with the grilled vegetables.
In a medium bowl, toss the arugula and radishes with the vinegar, the remaining 2 tablespoons of oil, and 1/4 teaspoon each salt and pepper. Drizzle the pizzas with additional oil and serve with the salad.