The catalog references the files by full path and file name. Since the drives are external, there is the possibility that the drive letters change as you connect and disconnect them. If that is the case, you can use the Windows Disk Manager to change the drive letters or change the names of the drives inside the LR Folders pane.
Possible problem #2 is that it is a file permissions problem. "you" on pc1 is not the same as "you" on pc2. Check the permissions and ownership of the directories and files on the external drives.
Thanks ManiacJoe! Changing the drive letter completely solved the issue. You mentioned in your reply changing the drive letters or drive names inside the LR Folders pane. I looked at the LR folders for the catalog to see if it indicated which drive letter was used while creating the catalog. I didn't see that information. Please advise.
That is what I get for winging it.
The drive element seems to be the one thing you cannot update/change. Instead, you would need to update the root folder under the drive to be on the external drive as its new letter.