To protect the wood, that's what they're there for. Wood Wailers are considered the guards and protectors of the Twelveswood. Yet one takes it upon herself to violently maim any cute little squirrels that approach. Cute, innocent squrrels.
Rise up fellow Eorzeans, rise up I say! Let us band together and stop this ill-tempered wailer from killing any more innocent creatures! Just because she's bored doesn't mean she has the right to kill our passive furry friends! Save the woods, save the squirrels!
Seriously though, devs you might want to have a look at this particular NPC. Whenever that one little squirrel spawns on the path this NPC goes nuts and chases after it to kill it, and then any that come to its aid. My battle log is being spammed by her whenever I come to the Treespeak Stables.