I gave up eggs on a trial basis to see if it would help my endometriosis.
What it did do was reverse the severe constipation I've had for years.
How can I know if this constipation is a simple constituency of stool problem, and therefore can eat small amounts of eggs below whatever threshold causes constipation for me. Or if it a sign of some true intolerance and I should avoid them completely.
Previous average consumption was an average of 2 per day, I would guess.
Food allergies can cause constipation. Usually due to the body's inflammatory response which can damage gut flora which, in turn, reduces absorption of antioxidants and vitamins/minerals.
A lot of people have allergies to egg whites, fewer to egg yolks. I would assume, given your statement, that you may have an egg allergy. If you can afford it, get tested. If not, experiment. Especially with egg yolks as they are very nutrient dense. So you could try to re-introduce egg yolks and see if there is any reaction.
source egg allergies: http://www.aafa.org/display.cfm?id=9&sub=20&cont=523
Have you tried other eggs besides chicken eggs? Many that react to chicken eggs are able to eat duck, quail, ostrich, etc., without problem. As CD suggested, you might react to the proteins in the white and not be bothered by the yolk. To answer your question about threshold, you will have to engage in self experimentation. I would start with just yolk and see if I suffered from the same issues.
You have to put some EFFORT into pooping, repeat the Mantra "I WILL LIVE OR DIE BY MY POOP!"
There can be no distractions, you must commit yourself to 5 years of hard, long training as a Poop Ninja, don't bother me with your excuses, until you have completed the training you will not be a true Poopster.