Will piece of Acrylic sag?
Commenters are right: Everything sags under its own weight. The question is how much you'll tolerate.

There are simple formulas for the deflection of simple structures like this. For a uniformly loaded square plate with simply supported edges,

.

And for a plate with clamped edges, 

,

where d is maximum deflection, W is the load, L is the length of a side, E is the elastic modulus and t is the thickness.

Your acrylic is uniformly loaded by its own weight. For now let's ignore extra deflection caused by poking fingers. Since you might attach your acrylic square to the rest of the table somehow, its actual deflection will probably be between these two numbers.

Let's do the worst case, where there is no overhang and you're supporting the acrylic right at the edges:

W = 1.18 g/cc * (36 in)^2 * 0.236 in * 9.81 m/s^2 = 58 N

L = 36 in = 0.9144 m

E = 2.6 GPa

t = 0.236 in = 6 mm

For supported edges d = 3.8 mm, for clamped edges d = 1.2 mm.

To a decent approximation, leaving a one-inch border just changes 36 in to 34 in:

W = 1.18 g/cc * (34 in)^2 * 0.236 in * 9.81 m/s^2 = 51.8 N

L = 34 in = 0.8636 m

For supported edges d = 3.0 mm, for clamped edges d = 0.9 mm.