The Joe Jost pickled eggs should be cured in the brine for two to three days.  After that put the sealed brining jar into the refrigerator to slow or halt the curing process.  The eggs should remain tasty for about two weeks although my pickled eggs don't last much more than a week especially when the kids and grand-kids stop by the house.  I usually make three dozen eggs at a time. 
 
Joe Jost puts them out onto the counter during business hours only and keeps them in the refrigerator when the bar is closed.  They go through so many eggs that a jar is on the counter for only a couple of hours.  Joe Jost also strains the brining liquid of the garlic and pickling spices after scooping the chili peppers out of the jar.  The brine liquid is reused to store the eggs and a bunch of the used chili peppers are replaced into the jar for serving with the eggs.  Nothing quite as good as watching a game on TV, having a Joe Jost Special (sausage on rye bread, dill pickle, swiss cheese optional,  and mustard), a couple of Joe's eggs with pretzel sticks and a COLD beer.