Pound the chicken lightly to flatten a bit. It’ll help cook the fillets faster. Then, sprinkle with a little salt and pepper.
Heat a non-stick pan (a cast iron pan would be even better). Arrange the chicken in a single layer, skin side down, and cook over medium-high heat until the skin is lightly crisp and browned. Flip to brown the opposite side.
While the chicken grills in the pan, cook the squash and chayote in about a half cup of water with some salt. Don’t use too much water. You just want the veggies to cook in the steam. Do this while the chicken cooks so that the chicken and the veggies will be done at about the same time.
Transfer the veggies with whatever little liquid is left to a bowl. Keep hot.
Cut the chicken into strips.
Heat the olive oil in a frying pan. Saute the onion and tomatoes for about a minute. Add the cooked vegetables (including any liquid), chicken and pesto. Toss lightly and cook just until everything is heated through and every piece of vegetable and chicken is coated with pesto. Taste, add more lemon juice, if you prefer. Additional salt and pepper might be called for as well.