The boss is selected randomly (most commonly Tanjis, Obujos and then Jaul) and remains for that day only. At server save the boss will be re-rolled and repeated bosses are possible.
If you do not open the guardian's chest after killing one of the Deepling guardians you will get the chance to fight the other two guardians on the same character the next days depending if they visit their lairs or not. That means if your are lucky you actually have the chance to kill all three guardians with one character while the third stage is open and then take all rewards from their chests within the order you killed them in. Note that this will only work as long as you do not take the reward from any chest before killing all three Deepling guardians.