When we first bought the house, there was no water softener. Our water comes from a well. The toilets had a reddish-brown stain which would start to appear after several days. We had the water tested which showed the iron and manganese content and got a quote for a water softener. I was going to wait to install one but when we hooked up our new washing machine, I ran it empty to make sure everything was OK. The wash tub was white and as the water was filling up I could start to see the reddish brown tint and see stuff floating in the water. I got the softener installed soon after.
Since having the water softener, there are no more issues with the reddish brown toilet staining and no more stuff floating in the water (it's been about 10 years now).
I checked the Culligan site and they say they can usually remove soluable iron and particulate iron if the particles are large enough.
I believe that the filter media is supposed to trap calcium and magnesium ions (iron too?) then when you backwash the filter, it exchanges the trapped ions for sodium from the salt in the brine tank.
I've seen concerns about using softened water but have not seen specifically what the potential problem is with using it.