They are both complete lies, I lived on a farm for a number of years and I have watched my dad shoot rabbits before when they have had miximatosis, an injured fox and on one occasion a Mink who had entangled itself on a piece of farming equipment and was strangling itself.
On the other hand, I start buying chocolate-covered marshmallow rabbits as soon as they're available, which at my supermarket is right after Valentine's Day.