Apparently Biden was mixing his metaphors. There are several movies in question, the 1949 John Wayne film “She Wore a Yellow Ribbon," the phrase “dog-faced soldiers” occurs. While the term “pony soldiers” occurs in the 1953 John Wayne movie “Hondo” and the 1952 Tyrone Power western “Pony Soldier.” This basically backs up Russ Tinder's answer. In the Power movie, a Pony Soldier is a Royal Canadian Mountie. Essentially, it appears to be a homespun way to jokingly call someone a liar concocted by Biden from memories of the movies mentioned above.