I haven't read Treasure Island (shame on me!), but Black Dog, of course, is the name of one of the characters. The speaker is saying Black Dog himself isn't here, but there may be a metaphorical black dog that the other person is carrying around. This would be some dark side to the person, some negative characteristic.
Winston Churchill suffered from depression, and this he called his "black dog", perhaps alluding to the Stevenson quote. Churchill's black dog was also referred to by the band The Manic Street Preachers, in Black Dog on my Shoulder, which is on their album This Is My Truth Now Tell Me Yours.