The java design of the "enhanced for loop" was to not expose the iterator to code, but the only way to safely remove an item is to access the iterator. So in this case you have to do it old school:
for(Iterator<String> i = names.iterator(); i.hasNext();) {
String name = i.next();
//Do Something
i.remove();
}
If in the real code the enhanced for loop is really worth it, then you could add the items to a temporary collection and call removeAll on the list after the loop.
EDIT (re addendum): No, changing the list in any way outside the iterator.remove() method while iterating will cause problems. The only way around this is to use a CopyOnWriteArrayList, but that is really intended for concurrency issues.
The cheapest (in terms of lines of code) way to remove duplicates is to dump the list into a LinkedHashSet (and then back into a List if you need). This preserves insertion order while removing duplicates.