If you bothered to look at the link you supplied you would notice that there are not any Verizon phones listed that will work. When I try to load the player from my phone it says that my phone is not supported. I would think it is more of a Verizon issue than a RIMM issue since BB phones from At@T and T-Mobile seem to work. --
It definitely looks like CDMA devices aren't supported (which would be the reason no VZW or Sprint branded BBs are listed). That seems odd to me, but I guess it could be VZW and Sprint blocking the app.
I don't understand why the app wouldn't install though, unless it's actually checking the branding version of your phone, not the OS version/hardware.