I had to check the word 'virile', I understand you meant masculine ;)
No, this is just a phrase and not a sentence, and any form of "swój" refers to the subject of the sentence. So if anything, it could only mean "its (own) dog", the dog belongs to itself. Which doesn't make any grammatical sense, as it's not like "I'm a modern independent dog" ;)
If you even could use it, it should be "swój" as this is the masculine singular version.
In Nominative, "swój" isn't almost used at all, and the situations when it is are outside the scope of this course - also, they're rather colloquial.
The plurals are "masculine personal" and "not masculine-personal". So while a dog is masculine, it is not a person, and therefore the right version (if it would be possible here) would be "swoje".