Second, the defining characteristic of "grey goo" is that it's of an artificial or synthetic origin, and hence, the "grey" part of the concept intimates technology, but not much else. The only other characteristic is that "grey goo" is typically depicted as a swarm of microscopic contraptions stuck on a runaway chain reaction, although the consumables that sustain the reaction can be anything at all. So, there's no real limit to the inputs or outputs of this hypothetical monstrosity, except that it's an all-consuming blob or nebulous swarm.
Third, you contradict yourself, since you ascribe microbes as grey goo, but then you tack on "non biological" sustenance as an afterthought. The microbial life at hand are not necessarily autotrophs or even consumers on non biological material, they are all heterotrophs made of biological material, eating each other. Furthermore the microbes are not "grey goo" because they are of natural origin and process.
In conclusion, grey goo is a thought experiment about a man made disaster or industrial accident, involving microscopic systems gone haywire. No other requirements are really specific to the idea.