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Run the ArcGIS ‘Polygon to Polyline’ tool (Toolbox > Data Management Tools > Features > Polygon To Line) with:  "Input Features" = (input polygon shapefile); "Output Feature Class" = (as desired); "Identify and store polygon neighboring information (optional)" selected; and all environment settings set as default.

Sample input parameters:

Sample output:

Run the Polygons TO Polylines Advanced tool

Select the 'OBJECTID' field for the 'Link field'.  Do not select 'Export points'.  With newer versions of ET GeoWizards it may be necessary to make and select a new item, with values equal to 'FID' for this purpose.