Gluteus Maximus
The gluteus maximus gives your butt its round shape. This is the biggest muscle of the entire body, and it possesses the most strength capability. There are actually two parts that make up the gluteus maximus, a top and a bottom part. The top portion, known as the superior head, makes up most the fibers of the muscle group. It mainly functions as a hip extensor. The lower part, called the inferior head, also works as a hip extensor and is heavily involved during hip adduction, a movement in which you bring your thighs close together. When you do the first phase of the squat, which is bending your knees and flexing your hips, your gluteus maximus lengthens due to this hip-flexing motion. You should feel a slight stretch in the muscle while you do this. Then, as you extend your hips back up to the upright position, your gluteus maximus shortens. So, you should feel it flexing during this phase.
Biceps Femoris
The biceps femoris is a pair of two muscles that are a part of the hamstrings complex. These two muscles are the biceps femoris short head and biceps femoris long head. They are both located in the outer back thigh region. Only the long head crosses the hip joint, enabling you to extend your hips. This muscle also crosses your knee, so it can work as a knee flexor, allowing you to bend your leg. As for the short head, it only crosses the knee joint, so it cannot assist when you extend your hips. It is important to note that when you are performing hip extension during the squat, you knees are in a bent position. This position shortens the biceps femoris, so the long head is slack. As a result, it can only slightly assist during hip extension, therefore forcing the gluteus maximus to work harder.
Semimembranosus and Semitendinosus
The other two muscles of the hamstrings are the semimembranosus and semitendinosus. These muscles are located in the inner back thigh region. Similar to the biceps femoris long head, these two muscles cross both the hip and knee joints, so they are activated during both hip extension and knee flexion. When you squat, your knees are being bent, so similar to the biceps femoris long head, these two muscles are also slack while during the upward phase of the squat. This means that their activation will be lessened during hip extension, thus forcing the gluteus maximus to do most of the work. Therefore, the hamstrings will only act as assisting muscle group to your butt while you extend your hips with your knees bent.
Adductor Magnus
The largest muscle of your inner thighs is the adductor magnus. There are two parts to this muscle: anterior head and posterior head. Both heads function as hip adductors. The posterior head, however, also works as a hip extensor. Thus, when you are squatting, you are recruiting this muscle to large degree.