Well, since it's all science fiction anyway, who knows? We are free to imagine anything we want, especially since Warp Drive itself is such a flagrant violator of what is conceivably plausible within the Laws of Physics, Nature etc.
Personally, I have not noticed this 'phenomenon' but I have been amused by the sweeping of the ring plane particles across our field of view (the TV window) as the camera pans around the planet. The same artistic trick is used when showing Saturn's rings close up in our own solar system.
My reaction to this is skepticism because the density of the ring particles is so low that a normal camera lens would just show a void. At least for the rings of Saturn, the particles are so small...a few meters according to microwave radio studies of their reflectivity... that you would PROBABLY not see many of them in a camera field of view without the aid of a telescope. The smaller particles have such low reflectivity that it would take more than a simple camera to reveal them.
As for the Voyager sequence showing the shadow, this is to my mind rather unlikely because assuming the particles are ice, they would not act like an optical mirror, or a single smooth sheet of ice, to show much of a footprint to a shadow. Only at very 'large' distances would there likely be enough ring particles underneath the shadow footprint to give you much of a dark spot, but at such a distance, the ship would be too small to see. This distance is related to the density of ring particles in the plane so that low density rings require you to be farther away, and high density rings, closer in. This would actually be an interesting calculation to do just to see what the ring conditions have to be in order to provide a recognizable shadow of an object at a given distance from the ring, and a viewing location at a specific distance from the body. Since I don't recall either of the Voyagers ever seeing shadows ( other than those caused by certain electrostatic phenomena acting on small dust grains in the ring), it would not be a calculation that would lead to an observable confirmation. Science really doesn't need non-falsifiable theories, so the calculation would be a dead end project.
My last caveat is that I have not done any detailed calculations to prove any of the above points, I am just using my intuition as an astronomer...Now, in the realm of imagination, who's to say that there might not be planets 'out there' whose rings are made differently, with the optical properties needed to do the Voyager shadow trick!