Breaking a goose neck can be made quite efficient if you only grab them very firmly by the head, then flip them in a couple of loops with a rotary motion of your wrists.
I am a turkey farmer, and was trained in this method by my mother at a very young age.
"Wringing" by gripping the body and head with two hands and twisting is hopelessly difficult and messy, and I won't do it. I have used my "grip and flip" method to cull many, many domestic turkeys and to finish hundreds of wild geese. It is quick, efficient and painless to both parties.
However, I do sometimes shoot geese in the head if I think there is a chance of them flying or putting up a fight.
In any case, use a method that kills quickly, efficiently and with minimal stress and pain for you and the goose!