Preheat a grill to medium. Place the chicken, breast side down, on a cutting board, with the large cavity facing you. Using kitchen scissors and starting at the open cavity, cut down each side of the backbone; discard the backbone. Open the chicken like a book, then flip over. Press down firmly on the top fleshy part of the breast until you hear the bone crack.
Season the chicken on both sides with salt and pepper. Grill skin side up, covered, until well browned, about 25 minutes. Turn and cook through, about 15 minutes.
Meanwhile, in a large bowl, toss the squash with 1 tbsp. EVOO; season with salt and pepper. Grill, turning, until tender and golden, about 7 minutes. In a small bowl, combine the lemon zest and juice, mint, almonds, garlic and remaining EVOO. To serve, drizzle the chicken and squash with the mint-almond pesto.