There are several possible explanations for the black satin underwear that contains a pattern of white whales with red eyes. The most obvious reference is to the white whale in Moby Dick. The white whale represents evil to the main character in Moby Dick and many suggest that the white whale represents Nurse Ratched, making McMurphy a representation of Captain Ahab who is obsessed with killing the white whale. That interpretation works to a point, but another explanation is that McMurphy is the white whale and Nurse Ratched is Captain Abab. this interpretation makes more sense to me, given the evil and obsessive nature of Nurse Ratched. A third interpretation associated with the underwear also comes from Moby Dick. Some associated the white whale with God, and since McMurphy becomes a Christ-figure who gives his life for Chief Bromden, a possible counterpart for the observer Ishmael in Moby Dick, this may also be a plausible symbolic explanation. It also may be true that Kesey is simply poking fun at critics who see symbols in everything, including the pattern of someone's underwear.
The tattoo is another allusion to a story that surrounds the death of Wild Bill Hickock. According to legend, Hickock was killed while playing cards. When he was shot, he was supposedly holding a hand that included two black eights and two black aces. Thus, aces and eights are known as the "dead man's hand". McMurphy's tattoo of "aces and eights" would obviously foreshadow his eventual demise.