I've used both eezox and CLP/Breakfree (the old formula).
As for lubricants, should you need a penetrating agent, I'd stick with the CLP. If you do the Eezox right it really works, but the time it takes to do it properly kills me. I'd rather sit down and clean and protect in one fail swoop.
I usually use the Eezox on the inside of the barrels of my pistols and rifles, the rails of my automatics, the ejectors of my wheelguns, and the exteriors overall if I use it at all. I try to use the CLP in instances where I don't have to worry about attracting fouling, dirt, dust, or brass.
As for cleaning, I just scrub and soak the parts in CLP, and use Brake cleaner to blast off.