I do not know if this is anyone that is related to your John Cook but I have a photo on my tree of a William O Cook who served in Germany with my Father in WW 1. If this is any relation I would be glad to share the photo with you. The men went from Germany/France to Camp Zachary Taylor, KY. At the time they were at Camp Taylor Pvt Cook was single. It is on the 1920 census for Camp Taylor. Hope this is of help to you or someone you may know.