As others have suggested, placing a piece of scrap underneath the hole will tend to prevent "blow out" on the back side of the piece.
You might also think about improving your technique, however. One thing I'd suggest is that putting a lot of pressure on the drill causes the wood to break when it gets thin. You might have better results if you ease up on the drill as it's about to emerge from the back of the piece. In addition, sharper bits tend to work better. MDF, in particular, is very hard on tools. If your forstner bit is dull, it will tend to rip, crush, and break wood fibers rather than cutting through them cleanly.
BTW, sometimes you simple can't get a scrap block in place as a backer. For instance, you might need to drill holes for new controls in an existing panel. An alternative to the backer block, which isn't quite as good, is to cover the back of the wood with duct tape. The idea here is that it'll hold the wood in place until the drill bit has a chance to cut through it cleanly. Again, you'll need to ease up on the pressure as the bit is about to exit the piece.