I am having the same issue since installing the recent patch. Selecting a server then attempting to logon locks up the browser. The only option is to exit to the main menu.
On an official server if you choose to backout instead of enter you are connected. There is no way to connect to a private server that I can figure out. I do play on a private server and have not been able to logon or see the password screen since the patch.
I have tried deleting and reinstalling the game without success. My PS4 is on the DMZ. I have tried to connect to alternate private servers to test and again no password screen.
Also I have not used any button remapping on my PS4.
Please help