WWII Jack is what Jack would have been like in the WWII era. He seems eager to defend his country, and regretful that Eric can't join up because of his mental illness, playing along that Eric was rejected because his ticklishness made him a torture risk.
When Eric gives Jack a picture of himself to keep him company in the foxhole, he tries to reject it, under the correct belief that the other soldiers would think he's gay. The concept totally flies over Eric's head, though, so Jack ends up bringing the picture. The other soldiers do think he's gay despite repeated denials.