Are you exporting to Fixed Layout EPUB (FXL) or reflowable? The latter is very much suggested.
That said, a document composed of separate text boxes is going to be difficult to manage in EPUB form, especially if you want a responsive display. You may be better off exporting to an HTML document with attention to responsive design.
The problem you will have, then, is that text boxes won't necessarily behave the way they do in print or FXL. Everything in an EPUB doc has to be one text flow, with any other elements (such as images) anchored within the flow. Simply putting elements on pages in the document won't necessarily give desired results. (Unanchored items "fall" to the end of the doc, not always in a predictable order.)
The other problem is that EPUB is highly dependent on the reader used, and they all vary. What you're seeing isn't as much the different platforms as it is different readers. Unless you can control which reader the (er, human) reader uses, it's hard to have precise control over the result. HTML and PDF are superior in that way, at least.
But yes, it can be done, with careful design to get consistent results on a vanilla reader like Thorium or Calibre.
Thanks. I appreciate your recommendations.