Yup,some time ago SCOTUS ruled that claiming military heroism/achievement simply to impress others is allowed by the First Amendment.
It was a good ruling, as that was one of the dumbest pieces of legislation ever to come out of Washington. You can't punish someone for lying about who they are unless they are using their fake identity to commit some other type of crime. You can't have a situation where I'm allowed to run around calling myself Donald Trump without any problem at all, but I can get prosecuted for falsely calling myself a former Green Beret.