A recent article in the New York Times addresses the concern that came to light following a cat catching H1N1 following his owner’s flu-symptoms. Although there has yet to be a dog diagnosed with H1N1, there have been confirmed cases of feline and swine H1n1 (Swine swine flu?).
What to do? Same thing you’d do to avoid inflicting your flu on your fellow humans – get rest, cover when you cough, wash your hands frequently, and avoid hugs and kisses until you are on-the-mend.