radiation pressure

Radiation pressure tend to be a weak "force" so I would question it likely used as a prime mover in compression.

However like everything else you need to run the numbers.

I suspect you run into two problems:

1) how to prevent the ablation, because the energy absorbed from the X-rays is going to heat up your material.
2) the force created by the pressure wave when the outer layer is vaporized is an order of magnitude greater than the pressure of the X-rays themselves.