A simple way would be to have your national borders in a separate file from your subnational borders.
Here is an example if you use the most-detailed border datasets from Natural Earth Data (http://www.naturalearthdata.com/downloads/10m-cultural-vectors/). I downloaded both the "Admin 0 - Countries" and "Admin 1- States/Provinces" datasets. They should align perfectly because Natural Earth Data is awesome like that.
I used a simple fill symbol with border for the Admin 1 layer.
I then stacked the Admin 0 layer on top using a transparent fill and a different border:
Because Natural Earth Data borders at the same scale line up perfectly, the national borders hide the subnational borders below them:
Alternatively, If you don't want the country borders to continue along the coastlines, you can download the "Admin 0 - Boundary Lines - land boundaries" layer for national boundaries instead: