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.