No, but they can be carriers of strep throat, MRSA, pink eye, ring worm, scabies, and mumps!
I know dogs and cats get a form of animal hep but don't believe it can cross over to humans. Had a cat in the 70s that got hepatitis and my mom nursed it back to health took about 5 months.
Not from you and not you from them...but seems dogs have their own version called hepacivirus. (And no, I'm not joking.) Interesting reading.
http://www.virology.ws/2011/05/25/canine-hepacivirus-a-relative-of-hepatitis-c-virus/
http://www.sciencedaily.com/releases/2011/05/110523152344.htm
How would they do that - it's transferred blood to blood so unless they are injecting drugs with you or something..........
And no they do not get HCV (unless you have a chimpanzee then it is possible but again if you aren't sharing needles with it or razors or something I can't see how it could get it).