Have you checked to see if they really are just using black? It's possible they have blank CMY channels.
Thanks for the quick response! I don't know the Corel product very well, nor
am I adept in Adobe Illustrator.
If I open the .eps file in Illustrator how can I view these extra channels
if they exist?
Illustrator is NOT an EPS editor. You'd be better of distilling it into a PDF.
Place the problem file in InDesign and then go to Window > Output > Separations Preview. You can turn off the black channel there and see if there is still artwork on the other channels.