How does Colonel Sherburn justify shooting Boggs?
I do not really think that Colonel Sherburn ever comes straight out and says why he has shot Boggs. Instead, we have to infer it from what he does say before and after. From that, we can infer that he shoots Boggs to prove he is a man -- he shot Boggs because Boggs insulted him and a real man does not stand for that.
We can see this first before he shoots Boggs. He tells Boggs that he will not stand for his insults past 1 PM. We can see it later when the men come to lynch Sherburn. He tells them that they are not men while he is.
Between these, the implication is that men fight back when they are insulted. Boggs has insulted him and questioned his bravery and since he is a man, Sherburn has killed Boggs.