OP, you asked what the muscles and joint actions of the squat are. Fortunately, the joint actions are a fairly brief spiel, but you've brought a dissertation upon yourself with regards to muscles used. Be prepared for a lot of big words that you probably won't understand and I'm probably too lazy to explain, and be horrified to know that this is me keeping it simple.
The primary joint movements of the squat are knee extension, hip extension and plantar flexion (ankle extension).
All squat variations use the quadriceps. It's impossible to actively extend the knee against resistance without using the quads. For this reason, squats are often regarded as a quadriceps exercise, although that seriously undersells them. I personally have found that the deeper my squats, and thus the greater the range of motion at the knee, the more work my quadriceps do, especially at the VMO, so I don't entirely agree with the recommendation to only go to 90degrees, although I see where the point is coming from.
All properly executed squats will involve the glutes a fair amount, too, because they are major hip extensors. Just how much glute work is required will depend on how deep you squat, how far back you squat, and stance; but if your glutes are asleep while you're squatting, something's going wrong.
Hamstring involvement varies from just about negligeable through to them being the prime movers of the lift. Hamstrings are major hip extensors, so you might expect them to be just as involved as the glutes by default, however they are a multi-joint muscle group, with their other main function being knee flexion (the opposite of knee extension). Since the knees go into flexion on the eccentric and extension of the concentric, as you descend into the squat the hamstrings receive tension from the hip movement and lose tension from the knee movement. If there is more movement at the hips than the knees, as is common in powerlifting squats, then there is likely to be a lot of hamstring work in the squat. If there is more movement at the knees than the hips, as is common in an Olympic style front squat, then hamstring work will be fairly minimal. A lot of lifters falsely claim that the deeper you squat, the more you'll use your glutes and hamstrings, but their involvement is not dependent on depth. Glute involvement is dependent on how far the knees are turned out and the ROM at the hips; hamstring involvement is dependent on the balance between ROM at the hip and at the knee.
The adductors, which are the inner thigh muscles, are involved more if the feet and knees are turned out, and less if the thighs are parallel to each other.
The calves, especially soleus and the deep plantar flexors (not so much gastrocnemius, which is the biggest, most visible calf muscle) are involved in stabilising the feet, and in any plantar flexion. Some people can squat with vertical (or near vertical) shin angles, which means next to no plantar flexion occurs when they squat, while Olympic lifters let their knees go well and truly past their toes, which means a lot of plantar flexion. On some occasions, I've felt front squats in my lower claves more than anywhere else, for this reason.
The dorsi flexors (the muscles of the front of the shin and top of the foot) are also involved in stabilising the foot/heel/ankle.
Moving up the body, there's a lot of trunk work involved in squatting. The spinal erectors keep the back neutral, which in turn prevents the hips from tucking under at the bottom of the squat, which both protects the hips and lower back from injury, and allows the glutes and hamstrings to be involved to their fullest. Keeping the spine neutral also helps to keep the bar (wherever it's placed/held) over your heels.
The abdominal muscles act to reinforce the spinal erectors, which in turn allows for the transfer of force from the legs up the trunk and to the bar. Weightlifting belts are used to give the abs something to contract against, enhancing this effect. Not something you need to worry about if you're asking what muscles and joints are involved in squatting, but something that does become relevant to those involved in strength sports.
The rhomboids, trapezius and deltoids are all used to varying degrees, depending on the kind of squat being used. The ways in which these muscles are used will vary greatly depending on what kind of squat you're performing. For a low bar back squat, all of these muscles will be contracted to create a shelf on the shoulders where the bar will sit. For an overhead squat, it's a very different story, and there's a different story again with front squats.
The last muscles I'll mention are the lats. In my experience, and this seems to fit with the anatomy, the lats don't do much in overhead squats or front squats. I haven't gotten much out of them in high bar back squats, either. In low bar back squats, however, I've experienced greater contractions in my lats than I tend to experience in actual lat exercises, like pull downs and low rows. In any case, lat involvement further reinforces and supports the trunk position created by the erectors.
And it's at this point that I'll stop waffling on about squats. For now...