I have 3 older siblings who graduated from USU and have a deep hatred for BYU as a school and as a sports program. Much of it has to do with the err of superiority that BYU fans display. Being a student at BYU myself, this is as much reality as it is perception. Students walk around thinking they are attending a superior school and their classes must be so much harder than they're friends and associates who chose other schools. I mean, how could someone possibly choose to NOT go to BYU, right?
I think their hatred stems from the fact that BYU and Utah bailed out of the WAC just as they were starting to make headway about getting into the WAC. USU has always been the underdog when dealing with D-1 Utah schools.
I think your Pres. Hinckley quote may be a little out of context. Perhaps the current stance on homosexuality would be better served with your quote.