Yes, that's basically right.
The gravitational way has a direction (say, z) in which its propagating. Within the plane perpendicular to that direction (x-y), a circular ring of particles will at at one moment experience squeezing in one direction (x) and stretching the perpendicular direction (y). As the wave passes through and you move from the peak of the wave to the trough, the directions reverse, so the first direction (x) stretches and the other direction (y) squeezes. By "stretching" and "squeezing" I mean instantaneous additional (positive and negative) acceleration on top of the (much, much larger) acceleration from the background gravitational field provided by the Earth.
Here's a visualization:
https://www.researchgate.net/publication/313828462/figure/fi...
Just as a child can swing their legs at the resonant frequency of a swing to pump up their sinusoidal amplitude, a very weak gravitational wave can pump up a ring oscillator if it's oscillating at the ring's resonant frequency.
Exactly square gravitational waves are of course not possible, just as for electromagnetic waves. (They would have infinite energy at the corners.) But in principle you could get a close approximation. However, spacetime is incredibly stiff, and I think all the known real-world sources produce pretty smooth waves. I presume most violent events are mergers of existing black holes, and essentially always result from a smooth in-spiral rather, say, a sharp collision event. This is what the "chirp" signal looks like to the LIGO detector:
https://www.youtube.com/watch?v=TWqhUANNFXw
The effects of a square wave would be roughly as you would expect: instead of smoothly pumping up an oscillator, it would give it a sharp kick, just as with electromagnetism.
