How can I prove I have a certain Facebook account by signature? I would like to know how to prove I have certain Facebook account (or that I can login to that account).
You have to show that you can use the account. You cannot prove it offline, without accessing the account (or storing public information there) or else anyone else could produce the same "proof".

Normally proof of ownership of a web page is done by the other party producing some random value and you publishing that value on the page. That would work here too. Alternatively, since Facebook accounts double as email, the other party could send a random secret by email to the @facebook.com address of that account, which the supposed account owner would have to find out. (Plain email is often not really a very secure transport, though.)

If you insisted on using asymmetric cryptography, you would have to publish a public key using that account. Then you could prove ownership of the corresponding private key by producing signatures using it. This would have the advantage of only needing to publish the key using the account once.