I have to disagree that the phrase ‘he’ll only fight me gun’ doesn’t make sense. It clearly means that if his brother joins up - the singer of the song would personally shoot him for doing so as he is trying with his banter and threats to keep his brother
out of the bloodshed.

I believe whilst having a universal application to just about any war with the ‘it will be all over by Christmas ‘ a common saying in both world wars one and two , that most likely it does have its roots in the Vietnam war where massive defeats ,losses and subsequent retreats were inflicted on US troops some of whom were Irish - American soldiers .