Hi Ryan,
Sorry about the delay in replying to this post. Everything that everyone has suggested here will work... but XPress has a much easier way to do this. In fact, this is an area of functionality that QuarkXPress is great for... asset conversion from print layouts.
The simplest way to do this would be to duplicate the layout as a web layout via the Layout properties dialog (Layout > Duplicate, then change the Layout type to Web). Then in the new layout, select the box you are concerned with and go into the Modify dialog... There is a checkbox option in there called "Convert to Graphic" which should be checked by default... make sure it's checked. Under the Export tab of the Modify dialog, choose JPEG as the "Export As" file type for that text box (we default to GIF or PNG for text).
From there, you can either preview the layout in a browser (Page > Preview HTML) and save the picture of the text (right click or control + click and use the save picture as option)... or if you want multiple boxes or long runs of text, simply export the page (File > Export > HTML) and in the Images folder that is created after the export, your text boxes should be one of the image assets in that folder.
If you exported, also notice that there are all of the other images from the layout in that folder, converted into web ready formats... so if you need to get image assets out of a print layout (say an eps or a tiff) you just do the same thing... duplicate the layout and export it in HTML... they are automatically converted for you. You can change the file format and even have some control over compression quality in the modify dialog for each picture.
I hope that saves you some time and effort!
Drew Bartlett (Quark)
Technical Product Management - Desktop