YYou will want to make sure that the layers you plan to merge have the same attribute table structure, and if not, you may need to create a script to update this. Also, .eoo files usually are coverages. coverages can contain arcs/lines, points, polygons (with a single point within containing the attribute information), tics (four or more extent points, or points that were used for paper digitizing), and one or more region features. these need to be merged separately.
hopefully they have given you some guidelines on which of these data types you need. They will also need to understand, that coverages had a "forced" Topology in place, and all these different feature types could be keep in one coverage, representing different, but related information. You lose that when moving to a shape file. Also, when merging, for example, typically it will keep all the polygons, even if overlapping (I say typically, since snapping distances can help or mess up things).
I did a similar exercise in 2010 when I finally moved our 27 coverages to ArcSDE since the topology rules were finally working for our state (there was a bug until version 10.0 for us). These 27 coverages represented three main themes, but since one of these themes had two or three different attribute data structures, I had to manipulate those too. After doing many tests, it came down to creating and maintaining three master (merged) files from the "region feature", then recreating the other arc/line features (which there is important information) from those.
Depending on how many files you received from you client, it may be worth doing in Python. My scripts were not written to be completely generic to share, but could be modified fairly easily, if needed, but I don't have access to my computer this week. Nice thing about using scripts is you can run them, review, delete results, repeat as many times as needed to get it right.
I'm curious why they specified shape file as output? I would recommend a file geodatabbases (if not ArcSDE) so you can take advantage or topology rules when editing. When merging, depending on the input data, the output can be pretty messy. So again, do some tests because you may need to talk with your client to make the output useful (of course, I have know clue what the data is, so it may not be an issue at all)
...oh, and all the layers that you merge will need to have the same spatial reference.
sorry to be so long winded...but for me, having been the creator of all the coverage data going in, it was a really fun project to get it to what we use now (I.e., three master files, that then recreate the 27 older separations, on the fly)
