If a square grid was used, there would be four possible openings. If a hexagonal grid was used, six. It would probably devolve into a mess if everyone used every side as a way out. I think if we just use our best judgement, a more freeform level would emerge, without a mess. Of course, a few people could create open areas consisting of more than one tile, not just corridors.
Seeing as to how doing animations for big boss baddies is out of the question in a static image, the gameplay would probably be something akin to "get all the coins without dying". I think the members here at pixelation could come up with some pretty fantastic obstacles. Building off of that, maybe each person gets a certain number of coins to put in their tile. Upgrades to player abilities would be cool but they would require more planning and wouldn't really fit into an exquisite corpse type setting. Static enemies in a tile would preferrably have 2 frame animations for their moving, like a walking goomba-type floor scrubber or a wing blur on an insect like enemy.
Things to be predetermined: Jump height, character height, number of "coins" per tile
A square grid lends better to a linear platformer, because there are less ways to branch out. But I don't think anyone wants to stray from the cute portmanteau "Hexquisite".