Most Kindle ebooks are in the AZW format, which is interoperable with the MOBI format. The Boox 60 can read DRM-free MOBI ebooks, but in any case Calibre can convert DRM-free MOBI to ePub - which essentially all non-Kindle E Ink readers can display.
The issue is DRM (encryption). Not all Kindle ebooks have DRM, but the titles from major publishers are DRM ridden. It is possible to strip the DRM from AZW ebooks - the easiest way is to use one of two methods designed for Kindle for PC ebooks. This is likely legal in the US, provided you only strip the DRM from ebooks you bought and only use then for your own reading.
Books from B&N can either be in ePub format or in eReader format. Most now seem to be in ePub format. They come with B&N's own DRM based on your credit card number as a password. This DRM can be removed, using the "ignoble" scripts. Once DRM-free any ePub-capable reader can be used, and Calibre can convert from DRM-free ePub to DRM-free MOBI for the Kindles.
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