I have the same problem as well with El Capitan, but it's not the first time. I've had the "cannot log into server" error before. I had it with Snow Leopard. Mavericks didn't have it. I skipped the Yosemite upgrade. Now it's back again with El Capitan, so this is NOT a new problem. Couldn't sign in with iTunes OR Apps Store. Furthermore, I didn't always have it with El Cap. When I first bought my MacBook Pro it was fine. But at some point, not sure when, I lost the ability to log into iCloud.
TEMPORARY SOLUTION:
The ONLY thing remotely resembling a solution I've found is just a bandaid, but it allowed me to connect to iCloud. Start up in Safe Mode (restart the computer holding down the SHIFT key until you start up). This turns everything off except your base system software. Open up iTunes/App Store, sign in, do your updates, downloads, installs or make your purchases. When you're through, restart the computer as you normally would. You will still have the same problem of not being able to sign in, but at least you got your housekeeping done.
This sucks, and I'm hating watching Tim Cook turn Apple, the greatest tech company in the world, into yet another greedy company that puts out crappy products for top dollar. I MISS STEVE JOBS!!!!!! And I hate what Tim Cook is doing to Apple's mobile products. If I'd wanted a Samsung, I would've bought one. Apple is supposed to innovate, not imitate.