It's simply getting EVERY Puzzle Monster in every area and all the Customization Items in all the areas too.
Go to your Profile and check the following.
At the top of the monsters it should say 101/101.
Then inside the profile near the bottom it should say something like the following:
Puzzle Pieces Found 4,297
Puzzle Monsters Captured 163
Customization Items Obtained 510

That Monsters number is important as is the puzzle pieces number. My CI number is higher than most since I have everything, most of which are not from the squares themselves and are thus optional. I can't tell you what your minimum number should be.

but you must have one area somewhere you missed. Check just outside the Center of the Earth zone. Check all 9 Maps (Main, Scientists, Underwater, Pyramid, Far Away Lands, Center of the Earth, Wonderous Lands, Pirate Island, Crypts) and also the Christmas one.

313 CI's are what come from all the 153 squares in the game. Just look at all maps, all squares on every map, and be sure that they are 100%. If your overall is not 100% then at least one square in incomplete.