Can't sign into iTunes on Mac 10.7.5
My answer:
Since you mention that you are running Mac OS X Lion (10.7.5), the issue could be that your operating system is no longer supported. Try to update your Mac and see if it works.
My experience:
I had a similar issue on Mac OS X Mavericks (10.9). Once I updated to Mac OS X El Capitan (10.11), the issue went away.