Lots of people use multiple products when cleaning for their different properties. Eezox is great for corrosion prevention, but it's not an oil or grease. CLP fills that role somewhat, but I still use a lube after the CLP such as GunButter or Tetra grease/oil. CLP performed pretty well, so if you are shooting/cleaning the gun a lot, CLP is probably all you need. For stored gun, I'm still going to Eezox after cleaning. You still need to lube in either case.
Eezox is a bit more expensive than CLP and harder to find, so I don't clean with it, but I'm sure you could, maybe. Note that it's a chlorinated solvent. Not sure if that could hurt any polymer parts. I thought people using brake cleaner usually use the non-chlorinated types?
Some people try to get by with just one product thinking they are lubing and preventing rust, but as you can see it's not always the case. I had heard Militec was not good at preventing rust, but people still use it baked on as a lube in dusty environments where oil or grease would collect too much crud. I use it sometimes on new guns, but was surprised how bad it performed on rust!