Symbolically hanging is quite humiliating. Historically it was used against thieves, traitors and others that a self-respecting man would not want to be associated with. After all, hangings were often public.

Cyanide pills are supposed to kill rather quickly and without much pain (though I am not sure as to how true this is). Hanging on the other hand could be quite painful if you did not die immediately. Taking a pill would be far less intimidating so fear may be a factor in some minds. More importantly think of the context in which these pills would be taken. If you were captured, the enemy may interrogate, imprison, or torture you. Thus a pill would be a quick way out and would be readily on hand whereas a rope would not.

As for shooting, I'd imagine it would be perceived more as a soldier's death than hanging. I am unfamiliar with specific accounts where men would request such deaths or what options they were given.
Perhaps someone else can weigh in on that regard.