You dumped out the water ten minutes ago. The pot is cooled off, and empty. It only had whole eggs in it, still in the shells. Now tell me, do you wash that pot? Get out the sponge and the soap and scrub it like usual? Or do you give it a quick rinse and slam it in the drying rack? No judgement here, only curiosity.
No washing, the boiling water sterilized anything nefarious on the eggs. I don’t wash my cast iron either, just rinse and kill it with fire on the stove.