A) Archaic (WWII-era) army slang for someone whose brand of shit was too small-minded and lame to be called bullshit or horseshit. This was typically an obnoxious, petty tyrant of an officer, particularly the kind who liked to push people around with made-up rules and regulations. Best exemplified in popular culture by the character of Frank Burns on M.A.S.H.
B) A coward; basically calling them chicken, with a 4-letter word tacked on for emphasis.
(Please note that these two definitions CAN overlap.)
B) "That's a good answer, Wykowski. It's a chickenshit answer, but it's a good one." - Christopher Walken as Sgt. Merwin J. Toomey, "Biloxi Blues," 1988
This definition is cited by at least two books by Steven E. Ambrose, "Citizen Soldiers" and "Band of Brothers."
Soldier 2: What for?
Soldier 1: Some rear-echelon type saw me and said that I wasn't maintaing a professional appearance. I can't help it that we've been living in foxholes for the last week and because of the snipers, I can't shave.
Soldier 2: That's chickenshit.