Is it possible to squat quicker than gravity?

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).

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.

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.