Earlier quoted context omitted.
Nope. Pushing on the end of the beam sends a pressure wave through the beam. The button gets pushed when the wave reaches the other end. Wave travels significantly slower than light.
If you push on the beam and the other end does not move right away doesn't that imply that the beam has compressed and is not rigid?
So, in the best case, this method of communication is equivalent to communicating with electromagnetic waves. And they have a speed limit: the speed of light :)
So this wouldn't be faster than light communication.