The application you linked uses a background file http://econym.org.uk/gmap/states.xml for the coordinates with the following content:
<states>
<state name="Wyoming" colour="#ff0088">
<point lat="41.0037" lng="-104.0556"/>
<point lat="44.9949" lng="-104.0584"/>
<point lat="44.9998" lng="-111.0539"/>
<point lat="40.9986" lng="-111.0457"/>
<point lat="41.0006" lng="-104.0556"/>
</state>
...
</states>
You can load the shapefiles from Natural Earth into QGIS, and save them as KML in EPSG:4326 (that's the degree format you want)
The border information inside the KML file looks like:
<Polygon><outerBoundaryIs><LinearRing><coordinates>166.318054,-78.314743 166.290253,-78.310562 166.142761,-78.278076 166.11972,-78.270294 166.098358,-78.255569 166.140533,-78.237808 166.160858,-78.232239 166.188873,-78.220291 166.196686,-78.213913 166.198303,-78.207245 166.188324,-78.173355 166.183899,-78.166397 166.165558,-78.160019 166.13974,-78.154724 166.112213,-78.150558 166.089447,-78.142517 166.072754,-78.135025 166.063873,-78.121124 166.065552,-78.114746 166.08136,-78.101959 166.105804,-78.098068 166.134735,-78.100845 166.279694,-78.115005 166.408051,-78.137802 166.60141,-78.166397 166.686127,-78.177505 166.752228,-78.20668 166.76944,-78.214447 166.77417,-78.221115 166.766663,-78.227524 166.74527,-78.239746 166.606689,-78.262222 166.506104,-78.27446 166.434692,-78.284454 166.414703,-78.290024 166.372528,-78.308075 166.352203,-78.313629 166.318054,-78.314743</coordinates></LinearRing></outerBoundaryIs></Polygon>
So you have to do a bit of text processing to get the coordinates from one format to the other.