I would not. zinc oxide is poisonous if ingested and not really for bloody wounds. its more for a rash or sunburn. and it's antibacterial properties are weak.
pet store employess frequently give bad advice. i had a sick duckling that died because of bad advice from pet store employee.
if you cant get vetericyn or blu-kote (another good one) you can use antibacterial ointment like neosporin or triple anti bacterial ointment from any place like walmart or a any pharmacy/drugstore has it. just do not use the kind that has pain relief added.