The direct object, that is, the person or thing that receives the action of the verb. In this case, He has...something. What does he have? Our dog. So, our dog, unseren Hund, is the direct object and is in the accusative case. Hund is a,masculine noun, so the article or determiner takes the -n/-en ending.
i believe unseren is for masculine. check out http://www.deutschakademie.com/blog/possessivartikel-im-akkusativ/
and it is akkusativ because of the verb hat