Peel the eggplant and cut it into 1/2-inch slices. Lay the slices in a large colander in a single layer, and lightly salt them. Turn them over and salt the other side. Set the colander on a plate, and set aside for about half an hour. This will draw bitterness from the eggplant. (You may skip this step if you don't find eggplant bitter.) After half an hour, rinse the eggplant and chop it into 1/2-inch cubes.
Heat a large non-stick saucepan, and add the onion and 2 tablespoons water. Cook, stirring, until the onion begins to brown, and then add the eggplant and a little more water. Continue to cook, stirring occasionally, for about 8 minutes, until the eggplant is mostly cooked. Add the squash, bell peppers, and garlic, and continue to cook and stir for 3 more minutes.
Add the remaining ingredients, except for fresh basil. Cover and cook on low until all the flavors have blended, about 10 minutes. Add the basil and cook for a couple more minutes.