Extract boundary from detailed layers in ArcGIS Desktop or QGIS?

You can just do a dissolve on the county layer (be sure to have just the counties you want to dissolve selected).

You can use GRASS module: v.dissolve You only need to import your data into GRASS. Here is a screen-cast of transferring your data to Grass from inside QGis.

Allow me to plug my own project... Boundary Generator will give you both external and internal boundaries of every polygon (as line features).

Internal boundaries are attributed with the FID for each of the two polygons that share that border. External boundaries should have an FID of zero for one of those two, so they are easy to select out of the full result.

The nice thing about it vs doing dissolves is that I've added a couple of knobs over precision so it can deal with not-quite-perfect data. (How far apart should two polygon borders be to be considered a shared border? How much angle deviation is required?)

It's still in alpha and it's been a while since I've tackled an update; I'd love to hear how well it works for you!