With a bit more research, I've found another option for travel to Europe on June 21. You can get a saver award on United from Neward to Duesseldorf, Germany. The flights are:

Newark--Edinburgh UA 36 lv 8:05 PM arr 7:55 AM next day
Edinburgh--Duesseldorf LH 3473 lv 12:05 PM arr 2:40 PM

There are no saver awards available from Duesseldorf the day you wish to return, but you could combine a flight to Dusseldorf with a return flight from Brussels, Geneva, Zurich or other cities have been recommended as having award availability.

Hope this helps.