Brad,
Is this a rental property? If so and you would like to R square the door, it will involve some stucco repair to properly waterproof and flash it.
Reinforcing the jam with some longer screws is not a bad idea (I actually do this on all doors I install) but make sure you put them where the jam is already shimmed between it and the studs or it can throw it all out of whack. You could also use longer screws for half the hinge screws too and then they'll be into the studs as well. Sheetrock screws don't have very good sheer value, I would rather use like a #8 or 10 gold screw.
How about some sliding latches on the inside of the door, top and bottom that go into the door jam, but sleeved into the studs so you aren't relying on the strength of the MDF door jamb. They could even then be padlocked. Following me there?
Of course some better door hardware like Josh's deadbolt idea and definitely the NRP hinges.