The comments in http://www.thingiverse.com/thing:106568 mention a few problems with that version, and I had similar problems when I printed one, so I imported it into openscad (you'll need to download those original files to use the openscad source included with this thing) and produced this new version.
Uses 1.75mm filament as the pin instead of printing one. Adds some ribs to the area that clamps the paper to hold it better. Removes a feathered edge at the bottom that was too thin to print well.
Print the 3 stl files. Use a small drill bit to enlarge the holes if necessary so 1.75mm filament will fit through them. Glue a length of 1.75mm filament to each washer, run the other end of the filament through the holes in the platen and the stands and glue the other end of the filament to the stand so the platen can rotate freely, but is held in place side to side.
Now you can run the paper between the stand and the platen and clip the platen down to hold the paper in place. The small ribs on the platen and stand will grab the paper so it won't slide around.