Well, maybe OP didn't mean equal by the means of equals() method.
Maybe OP meant equal defined as: for every entry in custom-object-set (name it entry) check if there is entry.toString() in the string-set. Maybe.
We can not know for sure until OP clarifies the requirements.
As for the actual comparison, Set.containsAll() method might be helpful (given that you just want to see that all elements in one set is contained in another) or simply equals() method (I haven't tried it for a set)
However, as others have pointed out it does not make sense to compare two objects if they represent something else.
Its like saying i want to compare Set<Zombies> with Set<Zebras>