Step 1Trim the squash, and cut into one-half inch slices. If using wide-body squashes such as big pattypans, cut them in half first. Zucchini should be simply sliced, unless they're gigantic garden zucchini. Pat chicken pieces dry, then sprinkle them generously with salt and pepper.
Step 2In a large, heavy casserole, heat the oil until it is hot but not quite smoking. Brown the chicken well on all sides, about 5 minutes, either in two batches or using an additional skillet. Remove to a bowl.
Step 3Pour out all but about 1 tablespoon of the oil from the casserole. Saute the onion, garlic, cumin and jalapeno in the oil over medium-low heat until they are soft and translucent, about 2 minutes. Add the vinegar, turn heat to high, and deglaze the pan, scraping up any browned bits from the bottom, until just about a teaspoon of liquid remains.
Step 4Add all the chicken except the breasts to the casserole, then add the squashes, corn and tomatoes. Cover and simmer over medium heat for 15 minutes. Add the cilantro; taste the sauce, adding salt if necessary. Add the chicken breasts, stir to combine, cover and simmer for another 20 minutes.
Step 5Spoon off as much fat as you can, adjust seasoning. Serve with rice and sprinkle with cilantro.