Combine the cheese and arugula in a bowl. Cut a deep 2-inch-wide pocket in the thickest part of each chicken breast with a paring knife. Stuff with the arugula mixture. Brush with olive oil and season with salt and the Cajun spice blend.
Preheat a grill to high and brush the grates with vegetable oil. Grill the chicken until blackened and a thermometer inserted into the thickest part registers 155 degrees F, 8 to 10 minutes per side. Transfer to a cutting board.
Meanwhile, heat 1 tablespoon olive oil in a skillet over high heat. Add the lima beans, squash and corn, season with salt and cook until the squash is just tender, 2 to 3 minutes. Add the tomatoes and cook 2 more minutes. Remove from the heat and stir in the lime juice. Slice the chicken and serve with the succotash