We'd be happy to help you with this. I have a couple ideas of what might be happening but it is hard to tell without more information about how you are accessing the site. Will you send in a email to us here: https://support.chess.com/customer/portal/emails/new and let us know as much as possible about this please?
i.e. Are you using a mobile device and the app? (If so, which version and which OS/iOS?)
Is this new? I've just made two moves by mistake because of this and it doesn't seem like there's any way to turn it off. On both occasions I didn't even realise I'd selected anything because I play by dragging my pieces which should NOT count as clicking on a square!