Hunting down the local overrun squirrel population for dinner? Seems fit only for an uneducated backwoods hillbilly at first glance... but this is exactly what is proposed in this New York times feature discussing invasivores. These invasivores, along with so-called locavores, look to those invasive species such as armadillos, lionfish, pigeons, and others that they can find and hunt locally.
Now I personally feel that my vegetarianism is enough change to my sustainable diet, but, for those who wish to eat meat... this doesn't seem so crazy. All that roadkill may as well not go to waste. And if people were to eat locally hunted food it would be that much less supply wasted raising live stock for human consumption.
Whatever helps. Just make sure before heading out with your shot gun that you research local pollutants that could potentially ruin your delectable grackle stir fry.