Assuming you're using v7, QuarkXpress does not support transparency on the outsides of EPS files where rasterised effects like drop shadows occur, as far I know. Although you can import layered TIFFs, even if the background layer is set to transparent in Photoshop, it will still come in as a white box, or even worse, with a clipped and opaque edge around the drop shadow.
If you have Xpress 7, you could do your drop shadow within Quark itself, or if you have v6 or below, do the composite using a work round by exporting your Quark background as an EPS and placing that into your EPS with the type. Similarly, you can do the same within Photoshop which at least will output reliably, albeit by rasterising everything. Type within Photoshop should be rasterised at a minimum of 600ppi (1200-2400ppi preferred) if it's going to press.
I would suggest work rounds with PDFs except Quark is notoriously fussy about PDF import at higher PDF versions, piece of poo that it can be.
QuarkXpress v8 (due out soon) should be an improvement with EPS and native AI handling, but that's cold comfort right now. Alternatively, depending on the scale of the project, lay it all out in Illustrator or InDesign if it's a single page. You can export what you have in Quark as an EPS and use that as a basis for continuing on with it in a different app.