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1.
An individual with an unbreakable habit.

Although this definition is based on a state supreme court case in which a dog was sentenced to death for raiding a chicken coup, the term is most commonly used as a metaphor.
You'll never get that egg-sucking dog Rush off his oxy.
by ambiguator October 18, 2011
 
2.
This literally refers to a seriously ugly or mangy mongrel. Said to be descrbed this way because of its non-doglike penchant for eating the insides of eggs.

Metaphorically, it can be applied to a person (almost always male) who is out-and-out riff-raff: one of the inbred corps.
Ole Billy Bob was just an egg-sucking dog.
by Duckbutt July 31, 2006