Could it be that the placed eps files are opened by someone else on the network the moment you open the InDesign file and get the missing links message?
I'm curious why you think the files being open would affect the path?
I'm not sure about that. Not at all.
It's more a faint memory:
A side effect, if you open a file on a server in PhotoShop and write it back after editing:
a totally new file will be written from memory while the old version is deleted. (If I recall that right.)
Could also be a problem with the server without opening a placed file somewhere else. I do not remember the specific circumstances (OS on the server side in conjunction with the network protocol), but I can remember someone already ran into a similar problem. If I have time I'll do a little research on that.
But first I have to hear a bit more about the server, its OS and the network protocol from the OP. And the exact version of the OS of the workstation.
There are some threads on the German-Swiss forum "HilfDirSelbst" that indicate something like that.
Especially the path name problem seems silmilar… (which has nothing to do with an opened file by eg. PhotoShop)
No solution nor deeper analyses in the two mentioned threads.
Possible but it doesn't seem to be the case. I have verified that the files linked in the .indd file I have open are not in use by other designers.