Has your boyfriend had a bad attendance record? Has he been absent from work alot? If he has, that can be reason enough to be written up. You're going to have to try to remember if he skipped out on work without calling in. These things matter. I don't think the supervisor would be "badgering" your bf unless these things were true.
Management HAS to have reliable people, especially when you're talking about drivers for disabled people. They cannot just "call in" or skip out whenever they please. They HAVE to be reliable! If your bf can't do this job then maybe he'd better find another one. You're right - times ARE hard -- but they are hard for management too.
From what you've said, I don't see where they are being unreasonable. It sounds like your bf needs to grow up. Sorry.

Hugs, Lee