I can add some additional information to what you and Rick Pitz have discussed but there is nothing definitive as to whether your father was aboard JOHN BANVARD when that ship arrived at Anzio.
A “Ship Search” of the excellent website ConvoyWeb (http://www.convoyweb.org.uk/hague/index.html) shows all of the voyages of JOHN BANVARD at http://www.convoyweb.org.uk/ports/index.html?search.php?vessel=JOHN%20BANVARD~armain. According to this listing, JOHN BANVARD departed Baltimore on November 20, 1943, likely without troops aboard. She made a quick run down the Chesapeake Bay (18 hours or so) to Hampton Roads, i.e., the Norfolk area. She likely embarked troops in Norfolk and departed Hampton Roads on November 24 in convoy UGS-25. The nominal destination of the convoy was Port Said, Egypt, but well before reaching that port, JOHN BANVARD was one of 16 ships that departed the convoy and steamed to Oran, Algeria, arriving there December 12, 1943.
JOHN BANVARD departed Oran on December 22, joining a passing east-bound convoy, UGS-26, and remained with that convoy for two days until reaching Bizerte, Tunisia, on December 24. She probably did not enter Bizerte but on the same date continued, either unescorted or in an unspecified convoy, to Naples, Italy, arriving there on December 26, 1943. The ship remained in Naples for the better part of a month, until departing on January 21, 1944, arriving off Anzio on January 22.
According to the website World War II Troopships, JOHN BANVARD carried troops of the 82nd Engineer Combat Battalion while in convoy UGS-25 (see http://ww2troopships.com/crossings/1943b.htm and scroll to 43-11-23). The 82nd disembarked at Oran and eventually was transported to Liverpool, England, arriving there January 20, 1944. Later the unit landed at Omaha Beach on the Normandy coast on June 16, 1944 (see http://www.82ndengineers.org/82nd-ourhistory.htm).
Presumably your father was not in the 82nd Engineer Combat Battalion because you know he went to Anzio and the 82nd did not, and therefore it is unlikely he was aboard JOHN BANVARD when she left Hampton Roads. This illustrates a point Rick Pitz made, namely that a military unit might move on different ships if spending time at an intermediate port. On the other hand, neither does it prove that he and his unit were NOT aboard JOHN BANVARD if his unit joined the ship when it left Oran or when it departed from Naples. So, to be short and simple but not definitive, your father may or may not have been aboard JOHN BANVARD when the ship arrived at Anzio.
If you know the more specific unit of the 5th Army to which your father was assigned (regiment, battalion, etc.), a Google search for that unit may be useful. There are many unit histories to be found on the Internet, some of which identify the ship(s) in which the unit traveled as, for example, the unit history of the 82nd Engineer Combat Battalion as noted above.
Good luck in your further research.
Ron Carlson, Webmaster
Armed Guard / Merchant Marine website