It can be any units, they just have to be consistent. The units of c imply the units used in the equation every time that length or time comes up. The units of m imply the units every time that mass comes up. Therefore, the units of E, which include all three of length, mass, and time, must match those used by m and c. As long as the units are consistent, the equation will work. For instance, as you say above, if you measure E in joules, m in kg, and c in m/s, things are consistent because a joule (kg m^2 s^-2) is based on kg, meters, and seconds.
If you used the old CGS (centimeter, gram, second) you would measure mass in grams, the speed of light in cm/s, and energy in ergs. You would get a different number of joules than ergs, but if you use the same mass (not number for m, 1kg=1000g) the energy would be the same.
If you wanted to use feet/sec for c and pounds mass (lbm) for m, things would work just as well, you just have to measure energy in (lbm ft^2 s^-2)
Lots of relativity work is done using a system where length and time are the same, c=1 (no units, since speed=length/time and in this system length=time so speed=length/length=1), and therefore the equation is just E=m, or in other words, energy and mass are the same things. This is the whole point of the equation, and it kinda gets lost with the c^2 term there.
