Arduino + Ethernet Shield = no-ssl support = no Firebase
Arduino + Ethernet Shield + FirebasePHP = Firebase!
You definitely need to make some changes to the script to make it work for your setup but this should get you started. i.e. your Firebase url, whitelist your server's IP address, etc.
Build using the Firebase PHP library from:
Base library: @ktamas77 Token auth: @craigrusso Update & Push method: @mintao