Can't login to itunes

You should tell us a whole lot more. Examples:

• What version of OS X are you using? (Current is 10.11.3)
• What version of iTunes are you using? (Current is 12.3.2)
• You say "... log in to iTunes ..." How exactly are you attempting to do that? What did you press before attempting to log in to get to that point? How did you respond to a login prompt?
• You say "... nothing happens ..." What exactly does that mean? Does that mean you entered some data that simply remains on the screen? Can you not remove or change what appears?
• You say "... click the button ..." What button? What exactly does it say? In what context?

You tried uninstalling iTunes? You should definitely not have done that.

I suggest therefore you re-install OS X El Capitan from the Mac's App Store. Then update all your software.

Having the exact same issue here.

I've also started having this issue - no ability to sign - in, I never get the prompt to enter apple id or pw.

On my system, I have Parallel's Desktop and i can get itunes for Windows to work through it on the same computer.

I've gone into my apple account and Deauthorized all computers, then try to reauthorize my mac - but still no ability to log in.

the sign in dialog box simply doesn't appear.

resolved it by reinstalling OSX El Capitan full