> problem is that when I try to open the mirror of the
> site from the external hard drive using a different
> coputer it no longer displays as a website and many
> of the files dont open at all.[...]



Important question: Does it open differently from the INTERNAL hard drive? 
Links can be absolute or relative and if they are set to absolute then you
need the same drive letter and folder structure for everything to look the
same in two different locations.

In other words, if it opens fine on C:\My Web Sites\MyMirror but opens badly
on D:\MyMirror then you'll be okay on a different computer as long as you copy
it to C:\My Web Sites\MyMirror again.

Hope that helps,
~bd