Yes, I do that regularly. We buy Mt Olive brand pickles and save the pickle juice. When there is enough juice to submerge 6 or 8 hard boiled eggs I put everything in a Mason jar and stick it in the refrigerator. Between 1 and 2 weeks the eggs are good to go. My wife prefers eggs in the juice from Claussen dill slices. Once in a while I will add hot sauce to the mix.
The flavor, although not as good as from scratch pickled eggs, is pretty good. After about 3 weeks in the juice the eggs start getting a bit rubbery but they taste good.