I believe Bob Taylor was pulling your leg a bit.
The U.S. Navy Armed Guard had no official logo or patch, so a U.S. Navy logo would be entirely proper and fitting.
The Armed Guard had a strictly unofficial, cartoonish patch; see http://commons.wikimedia.org/wiki/File:USNAG.JPG for an example. You may be able to obtain a patch from Project Liberty Ship, P.O. Box 25846, Highlandtown Station, Baltimore, MD 21224-0546. Send a e-mail to email@example.com to inquire about availability and price. (John W. Brown is the name of a ship, not the name of a person, so don't address your message to "Dear John.")
Full disclosure: I am a member of Project Liberty Ship and a volunteer crewman aboard SS JOHN W. BROWN, one of only two surviving, operational World War II Liberty ships.
Ron Carlson, Webmaster
Armed Guard / Merchant Marine website