Hello, my client is logged-out of client area when he click some link in client area. He tried several browsers. Is it some WHMCS issue?

From your description, it sounds like there is an issue with PHP sessions on the server. One way to confirm this is to check the "Remember me until I logout" box, as it then switches to using cookies instead. If it works, then it confirms it's sessions that are the problem. I would suggest contacting your hosting provider or a server administrator to review your server's PHP configuration, particular sessions support, and ensure they are configured correctly.