My dear friend, Mike Hogan directed me to the ff:
"This phrase usually appears in the form 'As screwed up as Hogan's Goat.' The Dictionary of American Regional English lists it. It connotes the idea of stinks like Hogan's goat. The phrase is attested to as early as c. 1942, but is probably considerably older.

"Hogan in this sense is believed to derive from R.F. Outcault's cartoon of that name, better known as The Yellow Kid (c.1895-1901). Also in use is Hogan's alley used to connote untidiness."

http:// R.F. Outcault

My Mom has used the phrase in this way rather than for drunk people or goats. In that case she'd make reference to my Uncle Bucky.
My neighbor has a billy who will bleat when a microphone is presented. My singing voice has been compared favorably to his.