The new Paperwhite Kindle will not work with epub, the formats it can use are .txt, .pdf, .mobi, .prc, and the various azw Amazon formats.
If the epub books you've got don't have DRM protection, then yes, you can convert them to mobi or azw3 (the new KF8 format for Amazon) with calibre and put them on a new paperwhite. If the epub books you've got do have DRM (ie. Adobe DRM if you've downloaded books from Kobo), then you'll need to get the third party tools from Apprentice Alf to take care of that before you can convert the format.
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