Preheat a panini style grill. Lightly grease with nonstick spray.
Mix together the pesto and mayo in a bowl until combined. Set aside.
Season the chicken well on both sides with salt and pepper.
Heat a grill pan over medium heat. Once hot, add the oil. Cook the chicken on both sides golden brown to an internal temperature of 160 degrees using a thermometer. Set aside to rest for 3 minutes before slicing
Spray the squash with nonstick spray and season on both sides with salt and pepper. Using the grill pan on medium heat cook on both sides until tender. Then cut into thirds or smaller pieces and set aside.
Spread the pesto mayo on the wrap. Layer with cheese, spinach, squash, chicken, and tomatoes seasoned with salt and pepper. Fold in the ends of the wrap and roll up tightly.
Place the wrap onto the panini grill and cook until golden brown and the cheese is melted.Alternatively, lightly grease a the grill pan on medium heat, place the wrap seam side down cook to golden brown then flip over and do the same to the other side. Cut in half and serve with pickles and chips.