In a 2012 Federal Court case, United States vs Alverez, the United States Supreme Court ruled against a similar law which made it illegal to wear military medals one had not earned:
" United States v. Alvarez, 567 U.S. ___ (2012), was a United States Supreme Court case in which the Court struck down the Stolen Valor Act, a federal law that criminalized false statements about having a military medal. The law had been passed as an effort to stem instances where people falsely claimed to have won the medal in an attempt to protect the "valor" of those who really had. While a 6-3 majority of the Supreme Court agreed that the law was unconstitutional under the First Amendment's free speech protections, it could not agree on a single rationale. Four justices concluded that a statement's falsity is not enough, by itself, to exclude speech from First Amendment protection. Another two justices concluded that while false statements were entitled to some protection, the Stolen Valor Act was invalid because it could have achieved its objectives in less restrictive ways."
If claiming to have won a military medal and thus a military citation is considered by the United States Supreme Court to be within the law then how could simply claiming to be a United States veteran possibly be illegal? It would appear Greensboro's panhandling ordinance is a lawsuit waiting to happen.
And no, I don't support people being allowed to lie about their military service. Not nary one bit. But when the City of Greensboro starts enforcing laws against lying then the entire Greensboro City Council is subject to be jailed.