Is there a basic Java Set implementation that does not permit nulls?

I would say use composition instead of inheritance... it might be more work but it'll be more stable in the face of any changes that Sun might make to the Collections Framework.

public class NoNullSet<E> implements Set<E>
{
   /** The set that is wrapped. */
   final private Set<E> wrappedSet = new HashSet<E>();

   public boolean add(E e)
   {
     if (e == null) 
       throw new IllegalArgumentException("You cannot add null to a NoNullSet");
     return wrappedSet.add(e);
   }

   public boolean addAll(Collection<? extends E> c)
   {
     for (E e : c) add(e);
   }

   public void clear()
   { wrappedSet.clear(); }

   public boolean contains(Object o)
   { return wrappedSet.contains(o); }

   ... wrap the rest of them ...
}

Note that this implementation does not depend on addAll calling add (which is an implementation detail and should not be used because it cannot be guaranteed to remain true in all Java releases).

Better than extending a particular implementation, you can easily write a proxy implementation of Set that checks for nulls. This analogous to Collections.checkedSet. Other than being applicable to any implementation, you can also be sure that you have overridden all applicable methods. Many flaws have been found by extending concrete collections which then have additional methods added in later versions.