How do I clean a pot after boiling eggs?
Put some water in the pot
Add vinegar or citric acid (whatever you can get your hands on more easily). (Sorry, I have no measurements, I always eyeball it. I guess up to 5% for vinegar is fine, or one or two teaspoons of citric acid depending on the pot-size...)  
Heat to a rolling boil... (do NOT put your head over the pot to see it it works, especially when using vinegar)
After it has boiled for a minute, turn off the heat and let things cool down.  
Rinse the pot
Enjoy the (hopefully) shiny pot :)