Euthanizing is not a very good thing to talk about, much less describe in a message, but if it has to be done, and you are not going to harvest the chicken and consume the meat, then i have a method that works relatively well, and does not require touching the bird you have made the decision to "put down" or cull . It is best to look at it from the standpoint that you are removing the unhealthy bird from your healthy flock.
Use a spade shovel, (the kind found at the local hardware store with a wooden or fiberglass handle) and dig a small hole the size of the the chicken you intend to place it in it. place the chick in the hole alive, and as the bird site in the hole, quickly use the shovel as a combination spear/ ax and thrust downward through the chicken into the hole. Often the shear weight of the steel blade on the shovel will sever the bird quickly.
If your shovel is not quite sharp enought to act as a cutting tool, then use it as a pounding tool. swing the shovel over your right shoulder, using the flat part of the shovel to "smack" the bird.
This works just as well as the severing method, and death comes in a swift quick manner. Once the job is performed, use the shovel for it's intended purpose, and bury the remains of the poor chick.