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.