I gather the eggs from the nesting boxes, with just plain water from the sink, I wash all the eggs off and set the on a dish towel (I have one specifically for my eggs, so it won’t be confused with the other towels) and when they are dry, I put them in a carton in the fridge.
I rinse them in the sink, if there's stuck dirt or fecal matter I lightly scrub it off. Dry with a specific towel and them put them in a carton.
If I'm holding eggs back for incubating I don't clean them at all. I'll flick off any fecal matter but that's it.
I generally put them straight in the fridge unwashed - if you maintain a clean nest box there shouldn't be poop or mud on the eggs. The only time I clean them is if I'm planning to hard boil, in which case I rinse under lukewarm water and scrub over the surface with a brush that's only used for eggs.
I only wash mine right before using them, unless they're really dirty. For a bit my run was very muddy (til I got 2 hay bales and it fixed the problem, save for where they scratch under it for bugs) and the birds tracked mud into it. And today I got an egg that must have been laid while asleep or something because it was next to the roost bar and that one is definitely getting washed!
I rinse mine under warm water when I go to cook them and cook my eggs thoroughly (either 10 full minutes hardboiled timer set at boil or when I do scrambled eggs I cook them til they're very slightly browned and completely dry no wet or gooey spots). I have to- I'm extremely susceptible to foodborne illness (gastroenteritis will land me in the hospital), and I also think raw or underdone (aka gooey) eggs are gross, to be honest. I also have a 6 and 9 year old so don't want them to get sick either, so all foods are well-done here. Still tasty though- not overcooked but thoroughly cooked