Bob - thanks for posting this, it's a very moving story.
I believe you misinterpreted the "don't tell anyone I'm here" quote. Here's the two relevant paragraphs:
He earned a chestful of medals, but Monti agonized over all the killing war required, his family said. He returned from Afghanistan in 2003 with a Bronze Star for valor, but his mother recalled: “He didn’t like talking about it. Most of the time he just liked to be left alone. He’d say, ‘Don’t tell anybody I am here.’ He wasn’t proud of it.’’
When he was pressed about how he earned it, Janet Monti said, he’d finally blurt out something like, “I had to kill someone’s brother, or father, or sister.’’
It reads to me that he was only saying he wasn't proud of his Bronze Star. I'm not sure what the "don't tell aybody I'm here" comment is, but I think it might be that he didn't want anybody to know he was home on leave...