First of all, EPS from Illustrator is absolutely archaic. Save the Illustrator as PDF/X-4 or High Quality Print and place that PDF into InDesign.
Secondly, better check what color space is used in Illustrator. If the Illustrator document is setup as RGB as opposed to CMYK, your black-only line art will be actually RGB black. There is nothing you can do about fixing that in InDesign.
Good point- they are indeed rich blacks. Thanks for the nudge, closer inspection in Illustrator allowed me to set the color to Grayscale mode, which was hidden in a tiny flyout menu, and set the colors there. The PDF from that is now just 100K. Taking it all the way through InDesign to final PDF, it is now also just 100K. Hooray!