Here are my thoughts (not knowledge, just thoughts):

If you wanted to suck an egg directly from the shell, that would imply (1) you were hungry (2) you didn't have a dish to put it in or a spoon to eat it with, and (3) you didn't have a fire to cook it on, or a pan to cook it in.

I would imagine the people most likely to suck an egg would be (a) a fugitive hiding in the woods, for example, an escaped prisoner or slave; (b) a soldier in the field; (c) a refugee from either a war or natural disaster. The egg, I imagine, would be either "found", confiscated, or stolen.

I have a hunch this expression goes back to the time when many survivors of the Civil War were still alive--people who remembered what it was like to suck eggs.