|South Carolina Senator Lindsey Graham|
Apparently, Zora Neale Hurston, an American author who portrayed racial struggles in the early-1900s American South, and who wrote more than 50 short stories, plays, and essays, used the expression in her story, "Sweat." As Hurston used the expression, the element of "unworthiness" seems definitely to be present. In the story, the woman using the expression is indicated as "hating her husband as much as a suck-egg dog."
Johnny Cash has a song about "egg-sucking dogs," and one interpretation is that calling someone a "suck-egg dog" is to claim either that he is "a man who induces women to cheat on their mates," or is "someone with an unbreakable habit."