I do, and I wipe them down on a regular basis, particlarly before I head to the field. I have only had a serious problem with it on my 10 guage. That said, I was out all day in freezing rain and snow during KY fall turkey season. I got home and there were little rust rings where droplets had been. My 1993 45-70 has had hints of rust coming through the blueing over the years, even after just sitting in my cabinet. I have seen several on used racks that had minor rust showing however and it is something I am careful to look for when looking at a used Handi. I recently passed up on a used 30-30 Handi for just that reason. You can see rust on many used guns, but it is usually associated with wear or scratches or something like that. Handi's seem more likely to rust right through the blueing. That is my experience anyway. Knowing this has not stopped me from buying them though, it is just something I accept and plan on dealing with. I have/had several Handis/Ultras (4570 with 223 barrel, 10 guage, 22 mag, 500 magnum, and I used to have a 12 guage Ultra Slug).
I am open to new ideas though, and I will check out some of the other preservatives out there. My old favorite is Rem-Oil. The success you have had with Eezox is convincing and I will give it a try.