Battlescreen not working in Battlefield 4
Solution 1:
It looks like this is occurring with a lot of people. EA has recently provided a help link for Battlescreen connectivity issues. It appears that Battlescreen uses some ports that may be completely blocked incoming/outgoing by your firewall or antivirus, but some of the suggestions they give lead me to believe that it was poorly designed from a connectivity perspective.
Under "Advanced", they provide useful information like flushing the DNS cache, UO trace, plugging directly into the modem, etc.
However it would be useful in providing in answer if you could tell us if you have access to Battlescreen on your computer when you are not running Battlefield 4 alongside.