With that placed eps in InDesign, turn on overprint preview within InDesign. Does the white areas appear?
Did you save the eps with a transparent tif preview?
Regardless, it is best to maintain transparency until it reaches the printer. So, save out from Illustrator as AI or PDF 1.4+ and use that in InDesign. (as you mentioned you did in the second have of the post)
If you want to know the exact cause of the white areas, you need to describe in detail the make up of the elements or send me a sample file and I will report back here the cause.
unisus (at) pacbell (dot) net
Note: If you choose not to send, no hard feelings. If you do send, post here that you did so as this is my spammed email account and I never automatically check it.