Hi, Prolagus. I hadn't seen the word before today. Poking around a bit, I find both OED2 and MW3 put the main entry under "zinke", but judging from searches on Google Books, "zink" is the commonest spelling (60%). "Zinke" is around 30%, "zinck" and "zincke" around 6%, "zinka", "zinc", and "cink" are rare, and "zinque" was restricted to a couple of occurrences in French.
As wind players adopted new instruments the wind instrument that engendered the greatest musical change was the zinck (the Italian term is cornetto). This was an instrument made of wood, slightly curved, and was played with a small cup-chaped mouthpiece. In the late middle ages this instrument was made in one basic size, approximately twenty-five inches long, and performed primarily in the soprano register. --Keith Polk, 1992, German Instrumental Music of the Late Middle Ages, p. 72