When I click on a reading in course reserves, I get a login screen.
You are seeing this because Blackboard has timed out. Refresh Blackboard, or log out of Blackboard and log back in. The error should be corrected and you should be able to access course readings. If you still have the problem, try clearing the cache in your browser. Here's a guide with instructions on how to clear the cache in any browser: https://kb.iu.edu/d/ahic.