"This is not a font, but rather a shape file" seems quite clear to me. Whether it is or is not used within a linetype definition is a secondary consideration.
The answer - as stated several times - is to contact the supplier of the drawing.
Printable View
It was obvious from the original post that we are dealing with a shape file, not a font file. The only way to get prompted about a missing font file, is if you do not have a FONTMAP file, and FONTALT is set to nothing. If these conditions are true, then you get the dialog shown below:
Notice the dialog is not asking for a specific "SHX" file, it's only asking for a font name from the list of fonts (either .shx in the support file search path, or .TTF as installed in Windows)
When there is a missing shape file, you are prompted with a standard Windows file selection dialog, to select the actual SHX file that is missing. This is what the original message said.
A little late, but here it is if you need it.
To clarify, it is indeed a complex linetype Shapefile.