I need to have a block of text where the box itself is flushed right, but the text inside the box is left-justified.
I don't think you can do what you're trying to do in the mobi format. You can use text-align left/center/right/justify and assign left-margins (which will be converted to blockquotes), but that's about it.
You may be able to fudge something up with tables, but tables aren't really recommended for layout purposes with the MOBI format. You could also create an image of left-justified text and then right-align that image (which won't resize when a user changes font sizes).
But I would just use a normal justified paragraph and assign a large left margin to push the text to the right. It's about the best you're going to be able to with MOBI.