Thanks guys, alot of good advice. What I ended up doing was mixing a big bucket with warm water and vinegar and let the filter do its thing for a couple of hours. Then after letting it sit for a little bit, I broke it down again and cleaned it out. Next, I filled up the bucket again with just warm water and let the filter run for a couple more hours. I took it apart once again and there seemed to be no salt residue anywhere in the filter.
One thing that I did use bleach on were the little ceramic bio filter rings. Those were pretty gross and I thought they could be reused with a good cleaning. I have rinsed them plenty of times so they should be ok to use now.
With the thorough cleaning that I did for the filter, I am really confident that there should not be any problems. Thanks guys!