Kindle - ugh.
The problems with the kindle format is, first, it is 2 formats. One works on gen1 and gen2 of the kindle, the eink versions, and the bigger kindle DX. The other, more modern, format, works on the newer kindle eink models and the android-based, color tablets. I suspect that the first format based heavily on the old mobipocket format, is going away as those earlier kindle models give up the ghost and are replaced.
The other big family of problems with the kindle relate to ebooks in general. Ebooks give a lot of control to the reader. There is also a pseudo-PDF ebook format that looks almost exactly the way you set it up, only it can be unreadable on a small smartphone screen - just like pdf.
Some css things will work on the newer kindle format and not on the older. Some things, like embedding fonts, add to the size of your ebook and reduce the money that comes to you. Also, embedded fonts can be turned on and off by the reader.
Indenting block quotes has problems on small screens. Boxing paragraphs might work on the newer kindle format, I don’t know for sure, but I would bet it does not work on the older format.
My advice to you is to experiment. Compile to epub, and use the free-from-amazon Kindle Previewer program to convert the epub to mobi format, then see what the book looks like in the Previewer’s settings, for eink Kindle, for iPhone/iPad, and for the Kindle Fire. You may be disappointed to see that some things work on one and not the other.
The reason to go epub first is so you can adjust the files in epub using Sigil. Sigil will let you alter the css stylesheets; then you can reconvert the epub to mobi and see how it looks.
Also, if you have one or more Kindle devices, email the mobi file to your kindle and see how it looks and works on the device itself.
My best guess is that italics will work best for you, even though reading through long passages of italics is not ideal.
One trick that should work on all Kindles is to break off the quotation by horizontal rules. But with long passages that span more pages than one, your readers can get lost.