In "An Occurrence at Owl Creek Bridge" by Ambrose Bierce, what is the attitude of the soldiers toward the hanging?

Ambrose Bierce depicts the somber, dignified attitude of the soldiers toward the hanging of Peyton Farquhar from the Owl Creek Bridge. The soldiers' formal, solemn positions and stately manner correspond to the seriousness of their business upon the bridge. Each sentinel at the end of the bridge is described as standing in a "support" position, which is a "formal and unnatural" stance, as they wait for Peyton Farquhar to be hanged. Bierce writes, "It did not appear to be the duty of these two men to know what was occurring at the center of the bridge," as they stand in a formal, dignified manner (1).
The other soldiers facing the bridge are then described as standing "stonily," and Bierce compares them to statues adorning the bridge. The captain is portrayed as standing with his arms folded in a silent demeanor as he watches his subordinates at work. Bierce continues to describe the formal, ceremonial setting of the hanging by writing,

"Death is a dignitary who when he comes announced is to be received with formal manifestations of respect, even by those most familiar with him. In the code of military etiquette silence and fixity are forms of deference" (3).

Overall, the soldiers treat Peyton's hanging with reverence and respect, as they stand in formal positions and do not acknowledge the serious proceedings at the center of the bridge.

I would describe the attitude of the Union soldiers present for the hanging of Peyton Farquhar as solemn and respectful.  
Readers are told two soldiers are stationed at opposite ends of the bridge. Both of them are at "support" positions. Readers are told the position is a "formal and unnatural position, enforcing an erect carriage of the body." It's a respectful and proper body position. If the soldiers didn't care about the hanging, or had a "business as usual" attitude, I imagine they might be sitting while having a cigarette.  
Bierce describes more soldiers than the two sentries, and those soldiers also display solemn, respectful silence at the events happening in front of them. The company of soldiers witnessing the hanging stands at "parade rest." It's a slightly more relaxed position than the other two soldiers, but still requires soldiers to stand still in an orderly fashion. We are also told the soldiers are "staring stonily, motionless." They are not cracking jokes or smiling. They take the hanging very seriously, and Bierce tells his readers why a few lines later.  

Death is a dignitary who when he comes announced is to be received with formal manifestations of respect, even by those most familiar with him. In the code of military etiquette silence and fixity are forms of deference.

Despite being soldiers who are used to witnessing death, the men still give Farquhar their complete attention, silence, and respect.