I’m sorry, but that makes no sense either. You’re talking about code fences now, and those are limited to a set of pre-defined languages (like HTML, CSS, JavaScript). And that’s not part of MD per se, but of prism.js (in the case of DT) or another highlighter.
You can’t re-purpose code fences for something like
```box-important
`whatever
```
since box-important is not a pre-defined language. Nor can you define it (without a serious amount of work, that is). And even if you could, it would go against semantic HTML, as code blocks are for code, not for arbitrary information. The best (I think) approach would probably be an aside element (instead of a div), since that is meant to stand out from the normal text flow.
I posted a simpler solution already that avoids all but one HTML element. Just make sure to keep all the empty lines in that example, they are crucial.