Is it possible to squat quicker than gravity? It's definitely possible to propel your body using muscle to move towards the ground quicker than gravity. 
But is it possible to squat quicker than gravity?
Assuming squatting from your knees and not you bending down from your back like a lever, and that you're not gripping the ground with your feet and pulling your self down. 
If you dropped a basketball from the same height as your head, 
would it be possible to squat below the basket ball before the basket ball hit the ground?
 A: Your head would be able to beat the basketball, but your center of mass wouldn't. The reason for this is that your head is subject to the downward force due to the rest of your body, but your body is only subject to the force due to gravity and $\mathbf{F}^\textrm{ext} = M\frac{d^2}{dt^2}\mathbf{R}$ where $\mathbf{R}$ is your center of mass (which follows from Newton's second law for a particle).
A: Yes, sort of, with qualifiers only. Otherwise, No.
To do a squat, you are in essence pulling your legs up slower than gravity is pulling your body down. To squat faster that gravity you first need to accelerate pulling your legs up faster than gravity is pulling your body down. This is possible, but then is where the qualifiers become necessary to actually "squat quicker than gravity."
In Kung Fu (I'm an instructor) there is a concept of dropping into a horse stance. Done correctly, with no jump/hop first, you are doing exactly as I described - pulling your legs up faster than gravity is pulling you down. As the speed of your body towards the ground is still limited by gravity, you briefly have pulled your feet into the air. For the Kung Fu application, this allows you to position your feet into a stance before you hit the floor.
For the main question, this introduces qualifier #1: do you consider it a squat if your body is in squatting position without being on the ground yet? If yes, then you achieved a squat faster than gravity, just not on the ground. Other answers here believe no, this would not count as you must reach the ground to be squatting.
Now for qualifier #2: do you need to be free standing, or can your feet be strapped down? I'm guessing this is what is meant by "gripping the ground" in the original post. If you can have your feet held down, then your leg muscles pulling you down can be done faster than free fall acceleration due to gravity.
Qualifier #3 is introduced by the original post as a test: do you equate squatting faster than the basketball as being sufficient? In reality, there is also air resistance. If your body has less air resistance than the basketball, then it is possible to achieve a squat on the ground, without gripping the floor, (infinitesimally) faster than the basketball would reach the same point as your head when squatting. This would beat the test, but still not technically faster than gravity.
Qualifier #4: can you weight down your shoes? This would change the equation which can be viewed in different ways. The center of mass of your body combined with the clothing would simply have a lower center of gravity and while above statements would apply normally, you are moving your body faster because you are actually starting from a lower center of gravity with less distance to go. If you treat the feet or legs as separate entities than the torso, then they can be pulling the body down faster than gravity without being pulled up into the air. If you weight them down enough that it is more than the body weight and you are not strong enough to lift the feet off the ground then any leg pulling action would only move the torso down and can go faster than gravity. The weighted shoes could be considered as "gripping" your feet and may violate the original question but it is not "gripping the ground" so may not, though with enough weight the result is effectively the same.
You cannot fall faster than gravity but you can accelerate using other forces. The question hinges on whether any other forces used still counts as a squat for this purpose.
A: Your muscles generate internal forces, which, by themselves, cannot develop any net force on your body, that would be added to the gravitational force and increase the acceleration of your COM beyond g. The interaction of your body with the ground cannot help either, since the normal force is pointing up.
A: It's possible if you simultaneously throw a second basketball (or any projectile really) in the air. The upwards force you're applying to the second basketball would be applied downwards to you by the ball, thus accelerating your squatting motion.
A: Adding to PiKindOfGuy's answer: if you are able to pull your feet fast enough, essentially you will be squatting in mid air for a fraction of a second until you actually hit the ground.
I'm not able if anyone has ever tried to do that though (without jumping :) ).
A: Acceleration due to gravity is -9.8 ms^2, and both you and the basketball start accelerating at the same rate, the only way to do this is to create some force which propels you downward and to ensure that you do not resist the acceleration of gravity. As stated above the best way to go about this is to take your legs off of the ground.
A: Imagine you and the basketball are in space, in free fall. Your body is straight. There is a basketball by your head. Your center of mass is near your hips.
Now you pull your legs up towards your chest. Your center of mass has not moved, but now it's near your belly.  There is less distance between your head and your COM, and your COM did not move, so your head must've moved down a little.  Meanwhile the basketball has not moved.
Now imagine this whole scenario taking place on the surface of the earth. You start in a standing position with the basketball by your head.  You pull your legs up so quickly that you are falling.  Your COM and the basketball's COM are now falling at the same rate. (If we ignore atmospheric drag.)  As in space, your head was pulled down when your legs were pulled up. So your head is a little below the basketball, until you hit the ground.
But it wouldn't be fair to call that a "squat".
For it to be a squat you'd have to pull up your legs slowly enough that your feet remain on the ground, and still control your balance, but still fast enough to keep your head below the basketball. 
A: Yes it would, and in fact if you make any effort at all, this is inevitable.
This answer doesn't disagree with or contradict the others.
Provided you squat quickly enough that the part of your body not dropping, i.e. the part below the knees, is not applying all of its weight to the ground, this means your top half is lifting it, and to do that it must accelerate downwards faster than gravity by pulling up on the rest.
But the above answers are right, only a part of your body can do so, and only insofar as the remainder (the part below the knees) does the opposite - effectively becomes weightless to the same extent.
It's worth mentioning that gravity doesn't fall at a certain speed, it accelerates at a certain rate so technically it is actually the acceleration of some part of your body which would exceed this rate of acceleration. This is actually guaranteed at least momentarily, if you make any effort whatsoever.
