The original term was literal. Sucking eggs (egg will break open spilling raw egg into mouth of sucker) is actually a (gross, but real) way to eat them.
As a curiosity, dogs (as well as other animals) will often steal eggs and consume them in this fashion from hen houses.
I don't use the phrase--it was never part of my childhood, teen, or adult experience.
Here's a pretty good site that is worth a read. Spend some time at the bottom half of the page. I would still have a hard time working it into a conversation....