Seeking polygon shapefile of countries, states and islands?

You can use the site Global Administrative Area. There, you can find surfacic shapefile for each country. Each country is divided according to its administrative divisions (ex: provinces). You can also download a polygon for the entire world.

http://www.gadm.org

You can use the site free Global Data from this site. If you are looking for these countries there are some that are not here but a few may meet your needs. https://dragons8mycat.wordpress.com/gis-data-sources/

Also, ESRI just has their own open data on their website here: http://opendata.arcgis.com/

You could use free and open OpenStreetMap data which is licensed under Open Database License (ODbL).

Keep in mind that OpenStreetMap is a community driven project so the data might in some cases be incomplete.

Mapzen offered a service where you could download borders from every administrative unit within a country, as a GeoJSON, file but that service has been discontinued.