What is the ruling on compelling someone to swear by Allah by saying, "I adjure you by the Almighty Compeller to do such and such," or a similar phrase?

It is permissible to swear by the names and attributes of Allah, but not by anything else. A Muslim should not frequently swear oaths. The one for whom an oath is sworn (the "sworn-upon") is not obligated to fulfill the oath if it entails hardship, harm, or a contravention of Islamic law, though it is commendable for them to fulfill it. If the sworn-upon breaks the oath and does not fulfill it, the expiation is on the one who swore the oath, not on the one for whom it was sworn.