I speak as though you're misstating the claim, because you are. If you can refute the claim you will show it false, but if one is unable to refute the claim, nothing whatsoever is established about its truth or falsity. The claim is that you cannot affirmatively prove a negative - you have pointed out that you can disprove a negative. Wonderful, but that's not the claim advanced.
First, it gets most of its force from our a priori presumption that green unicorns really don't exist -- it verges on the fallacy of affirming the consequent.
Actually, no - there is no a priori assumption about the existence or nonexistence of unicorns, merely the observation that nobody has ever observed one. Hence, if they exist, it is not likely that they exist in places humans frequent. Additionally, affirming the consequent is a formal error of logic. There is no "verges on" committing such a formal fallacy - either one does or does not commit it, and in this example, there is no such fallacy.
We'd be far less impressed by your "universal, omniscient search" if we knew exactly where to produce the green unicorn that would refute your claim to the contrary.
In other words, if you know in advance that the claim is false, you'll find it easy to refute. Pardon me if I'm not impressed ;)
But I also noted that you can prove a negative, and gave a) a specific means to do so; and b) an admittedly poorly recollected example where it was actually done. (I could dig among my old texts if you really want chapter and verse.)
Thus it is demonstrably false that "you can't prove a negative."