I have tried my reloads in 3 different 870's and none of them want to eject them without really having to crank on the pump.  My buddy's Benelli eats them no problem along with a couple of my Non-Remington shotguns.  I just have a Mec Jr but I'm thinking about switching to the Mec that resizes the shell.  I've tried using AA's, Federal value packs, and Estate.  

Anyone else have this problem?

If not, what reloader and shells do you use?

I don't have any problems with reloads in my 12 gauge 870's, and haven't started loading yet for the 16 or the 20. My press is a 40+ year old Texan and has a resizer.

IMO, if you're not shooting the same shells in the same gun, you need to resize them.

No problems here.
I use a MEC 600 Jr. Mk5 loader with Winchester A-A cases for all of my 12 Ga. handloads. It sounds like your cases are not being sized enough. My ammo works in every shotgun I have tried them in.
Take one of your handloads, and drop it in the chamber of your shotgun. Does it drop in fully without help? If so, the problem isn't with your press. Its with your loads.

If they do drop in without help, are you thinking they are too hot of loads and expanding the  case too much?  I'm within the range according to the Hodgon website.  

How does the chamber look? It might be dirty, have tool marks, or pitting. Any of those can cause exactly the problem you're describing. If it's got tool marks or pitting, you'll need to get a proper flexhone and polish it up.

There are some specialized chamber brushes, but I don't use them. I take an undersized brush(like 20 gauge for a 12 gauge) and wrap it with 0000 steel wool. Then I screw it onto a cleaning rod section and chuck it up in a variable speed drill. Squirt WD40/RemOil/CLP/etc. on the brush/steel wool and into the chamber. Insert the brush into the chamber and use low speed while moving the brush in/out of the chamber. Keep it wet with oil--you'll probably want to put a bucket or catch pan under the muzzle or do all this outside. After a couple minutes of running the brush in/out, remove the brush and clean everything up. If a dirty chamber was the problem, it will run like a champ now.