There aren't a whole lot of restrictions on puzzles but a few of them are:
1) The cube must be stickerless or a single color of plastic with stickers - i.e. no carbon fiber cubes
2) The sides of the puzzle must be able to be clearly distinguished from each other - i.e. no two red sides
3) Can only only have 1 logo sticker and, if the puzzle has centers, it must be on a center
4) Can't have damage that distinguishes one from side from another by feel - i.e. no shredded stickers or big scratches or pieces half-popped
5) There are no size restrictions but you can't use a puzzle that is clearly intended to be a joke - i.e. no 15mm or 180mm cubes