Same issue here, pretty much when I open Evernote on iOS it asks me to sign into iTunes and the web clipper in Chrome keeps logging me out as well (I think it happens every time I close Chrome).
In the case of iOS I already deleted and installed it back because when Evernote 8 was out the app wouldn't open. In Chrome, I also removed the extension, installed it back but still the same.
I'm guessing that there is a connectiong between the iTunes sign in on iOS and the clipper in Chrome (I'm currently using it in Windows).
Since this seems quite widespread, and I didn't have this issue before I'm more inclined to believe that it has to do with the latest version of the app and not my connection, firewall, etc.