I am reminded of a cartoon I once saw, of a man with mustache and a mane of grey hair standing in front of a blackboard. On the board is written E=mc^4. It is crossed out. Underneath is written E=mc^3. It is also crossed out. And the man is pondering. I also recall a (true) story of the confusing days following the introduction of metric units in Canada, when an elderly lady wandered into a vegetable market in Winnipeg and asked for a kilometre of carrots. The man at the blackboard confronts a very similar problem.
Energy first entered physics in the guise of the kinetic energy of a falling body, E=mv^2 /2. Later it was gradually realized that energy can take many other forms, but this formula established the essential nature of what energy is, by specifying its physical dimensions, i.e., the units in which it must be measured: (mass) (length/time)^2. Just as a mass measured in kilometres is not really mass at all but something else, so E measured in other units would no longer be energy. So the man at the blackboard has no choice but c squared.