Preheat the oven to 400 degrees F. In a medium bowl, whisk together the ricotta, eggs, salt and pepper. Add the spinach and stir to combine. In a separate bowl, combine the mozzarella, Parmesan and basil. Heat a large skillet over medium-high heat and add the oil. Add the garlic and onions and cook, stirring, 1 minute. Add the carrots, yellow squash and zucchini, and cook until the vegetables are tender, 3 to 5 minutes. Season with salt and pepper. Remove from the heat and drain off any excess liquid. Spread a third of the marinara sauce over the bottom of a 9 by 13-inch baking dish. Arrange a layer of lasagna noodles on top. Spread a third of the ricotta mixture over the noodles, followed by a third of the cooked vegetables. Sprinkle with a third of the shredded cheese mixture. Repeat to make two more layers, ending with the shredded cheese mixture on top. Cover the top of the dish with aluminum foil and bake 30 minutes. Remove the foil and continue to bake until the top is golden brown, about 15 minutes more. Let cool at least 10 minutes before serving.