There's a reason he was the only one. The U.S. is pretty reluctant to execute someone for just being a coward. Almost everyone in Eddie's situation, including Eddie, was generally given multiple chances to avoid the ultimate penalty. For one thing, if someone is so much of a coward they'd be useless in combat anyway, why waste a gun on them? Almost everyone was given a chance to rejoin their unit, turn into an REMF, do something other than combat or, in only the most severe cases, do some time.
Ironically, for being a willful coward, he voluntarily put himself in front of a firing squad.