It is my guess that regardless of having a reef, overcrowding and the resulting stress is what is causing your Ick problem (if it wasn't obviously introduced that is through lack of QT, etc.). I don't believe even an army of cleaner shrimp will help, as some fish simply won't go to them, and I believe they are more of a hopeful measure to keep parasites away, than a legitimate treatment.
Between the Rock Beauty, Hippo Tang, and the Yellow Tang, I don't think there is enough swimming room to not cause stressful interactions. The RB and HT should really be in larger tanks, and depending on the dimensions of the 66g perhaps even the YT should too (most agree that 4' of swimming length is at least a very basic judgement of space required).
If you really have Ick, the only solution I see is to isolate all the fish separately and treat appropriately, including finding new homes for some of the fish before aggression becomes an issue as well.