Yes you can reuse pickled egg juice and you can put hard boiled eggs into the pickled egg juice and allow it to sit overnight or longer to help the eggs absorb the pickled egg juice.
Then the eggs will take on the flavor of the pickled egg juice and you'll have pickled eggs.
However for the best pickled eggs it's best to start off with fresh pickled egg juice and allow the eggs to sit in the brine for the correct amount of time.
But you can certainly hard boil some eggs and then remove the shell from them.
Then place them in the pickled egg juice you already have.
The eggs will be pickled taste at least somewhat.