Bullboxer staffy bull dog breed information and pictures, Bullboxer staffy bull information and pictures. the bullboxer staffy bull is not a purebred dog. it is a cross between the boxer and the staffordshire bull terrier.. Bullboxer dog breed information, The bull-boxer types go as far back as the early 1950s or more. many variations emerged and there was no real standard on just what a bullboxer really was. when a.