Import the AI version, not the EPS. The shadow is a bitmap effect and there will be an opaque background around the shadowed area.
Take care, Mike
Oh, alternatively, bring in the file (AI or EPS) with no drop shadow and use the Effects panel in ID to add the shadow.
EPS means Encapsulated PostScript (it's really PostScript code wrapped up in a capsule that allows you to place it into other applications as an image), and the PostScript specification doesn't include transparency. That's why even though you have transparency in the Illustrator image, it's format won't translate that transparency data into usable transparency when you place it into InDesign. Lack of transparency support is one of the main reasons people recommend not using eps in a modern workflow.