For self-defense Jews are allowed to own guns according to the laws of the land in which they live. Depending on the circumstances it might possibly be a need - such as in a border town in Israel.
Regarding hunting - for sport - there is discussion whether it is prohibitted according to Jewish Law, and if so, if the prohibition is of Torah or Rabbinical origin. Whatever the legal decision, the great Rabbi - Rabbi Landau, the Noda B'Yehuda - writes that it is definitely not a Jewish trait or characteristic to hunt animals and should not be done.
