As a professional mechanic we always referred to that part as the "caliper frame" and often cursed when we had to use a file to remove the rust behind the clip before we could re-install the clip and the pads - when a bit of grease or anti-seize when it was assembled last time would have prevented the problem
Smart idea. Coastal california vehicles should be lubed too - as should anything that gets onto the dry lakebeds or salt-flats. Or any vehicle in Georgia , Florida, or Louisiana.
A little bit of grease/antiseize won't HURT, regardless where you are.
You don'tn thow on a fist-full - just a light swipe or smudge is all that is required.