I have an .eps image that supposedly has no background, but is showing the gray and white checks no matter what I do. What's going on?
The grey and white checks that you are referring to are the way that the Adobe products show transparency. In other words, any areas of an image that have an overlay of grey and white are either completely or partially transparent.
The simple answer to your question is to view your document using Preview mode (the shortcut key to switch back and forth from preview mode is W--should work in all cases as long as you don't have your text tool selected.)
This might be a locked or hidden layer within the .eps file. Some stock images actually have a layer which replicates that checkered background which usually indicates transparency. Open the eps in illustrator. Check all the layers for some checkered background, unlock it, change its visibility... update link in indesign.