I frequently use Rem oil on my Glocks and I use it on my 870. Works fine on those. I've used it on the AR's, too, but mostly Lucas gun oil, CLP, or Mobil 1 5w/20, which really works well and doesn't burn off.
I could get by with just Rem oil, if I had to. I've got guns that my Dad handed down to me that have had nothing but Rem Oil and they are just fine with no rust. One is a shot gun purchased for him back around 1938. I don't know what he used back then, but he always said he used Remington oil. How similar it was to what is available now is questionable, but I'm 48 and as far back as I can remember, he lubed his guns after cleaning with Rem Oil. That old H&R shot gun still looks great. I've also got an early 60's Winchester .22 semi-auto that has had mostly Rem-oil its entire life and it is in perfect condition.
The Rem Oil I mostly use now on the bores of my guns for storage is from a spray bottle that I've only found at Wal Mart. It is marketed as being a strong rust inhibitor and it has worked fine for me for the last 4 or 5 years. No rust on any of my guns, and I've not had any lube related malfunctions, either.
Clean your guns frequently and properly and then lube them appropriately, whether for storage or use, and your guns should be fine, with Rem Oil or other mass marketed gun oils. I've got the proof in my safe. (I do like a bit thicker oils for the AR's, though, if working them hard. That's why I've used the Mobil 1 or Lucas gun oil, with Lucas being the current favorite, thanks to the AR being a much more harsh environment for gun lube.)