What router settings would prevent me from accessing CSS files via https?

Skype's static resources are hosted at Akamai, a content delivery network, that routes requests to one of their worldwide servers that is best suited for your location. Maybe the Akamai-generated Skype certificate is not valid on one of Akamai's servers, or maybe one of their servers simply does not host the Skype content at all (anymore) and Akamai is trying to show you an error page, using a different certificate.

So, just a wild guess: maybe the router uses different DNS settings, which are either outdated (maybe referring to a wrong IP address altogether), or referring to an Akamai server that is configured wrongly, or for filtering DNS servers (did you set up something like OpenDNS some day?) referring to an an access denied page of the DNS server.

You can try nslookup secure.skypeassets.com to try to find differences for the two connections. (The first line will tell you which DNS server is used, but that usually refers to the router or modem's IP address.)

You can also compare the output you get with the output when using Google's DNS:

nslookup secure.skypeassets.com 8.8.8.8

Server:     8.8.8.8
Address:    8.8.8.8#53

Non-authoritative answer:
secure.skypeassets.com  canonical name = secure.skypeassets.com.edgekey.net.
secure.skypeassets.com.edgekey.net  canonical name = secure.skypeassets.com.edgekey.net.globalredir.akadns.net.
secure.skypeassets.com.edgekey.net.globalredir.akadns.net   canonical name = e7766.b.akamaiedge.net.
Name:   e7766.b.akamaiedge.net
Address: 23.206.91.11

The IP address in the last line can surely be different without causing problems, but maybe you happen to find an IP address that should not be used.

It is possible that someone is running some sort of man in the middle attack. Login to the router and check to see who is connected. If you find some suspicious machine just change your wifi password and make sure you use WPA encryption