To wring is to twist, squeeze, or clasp firmly, especially to extract liquid. It is the appropriate word in wring [one’s] neck, meaning to choke. Ring would almost make sense there, as wringing a neck involves holding one’s fingers in a ringlike position. Nonetheless, wring is the conventional spelling in the phrase.
Most people raise a rooster until he is six or seven months old, then they take him out behind the house to wring his neck or take a hatchet and cut his head off. [Wetumpka Herald]
They will wring their necks or stuff them still alive into barrels to die from suffocation. [The Moderate Voice]