I think it would work for all finite fields.
One of the solutions to this IOI problem was solved that way, and it's not of the form GF(p) but it's GF(p^2).
http://olympiads.win.tue.nl/ioi/ioi94/contest/day2prb1/probl...
see solution 2:
http://olympiads.win.tue.nl/ioi/ioi94/contest/day2prb1/solut...
Good point. So it would seem to be hard to find solutions for 6, 10, 12, etc. colors. Have you given any thought to how to solve those cases?
Wikipedia has an article on this, http://en.wikipedia.org/wiki/Linear_equation_over_a_ring it would take a long time to read up the background to understand it, and I don't have the background to distill it to the exact result we want.
Note the wiki article references this book: Ideals, Varieties, and Algorithms by Cox, Little and O'Shea. This is a standard and approachable reference on this kind of topics.
Although this topic is interesting, I think it's best to treating it as a technology. Knowing it's existence and know when to apply them is good enough. Understanding how it works would be a big time sink and sadly doesn't give much useful insights.
Almost nothing about the solver would need to change in order to support computation over GF(p), which suggests that someone should create a version of Lights Out / Inverter which cycles through a prime number of colors rather than just off/on.