I don't have any proxy/port set there and I don't use IE as my default browser, I use Firefox.
Given the troubleshooting screen, I assume that (unless set up differently in Messenger itself) Yahoo! Messenger uses the same settings as IE. So you might need to change these settings even though for browsing you don't use it. So you'll need to figure out if the VPN you're using provides such proxy server, and then make IE or Messenger use it.
See also Yahoo's Troubleshooting Tips for Messenger Sign-in Issues. If your connection to the web is through a HTTP proxy, then you won't be able to make or receive voice calls with your PC, according to What ports are used for calling with Yahoo! Messenger? But I assume sending text messages should always work.
Note that the details show the problem:
Resolving host name vcs1.msg.yahoo.com
This should yield
188.8.131.52, but you get:
Connecting to Virtual IP server 127.0.1.5
127.0.1.5 is an IP address for a private network, not for the internet. So I assume that the VPN service you use also uses its own DNS server, to relay all your requests through their own web servers. That DNS server then does not give your browser or Messenger the true IP address, but a private address. This way they can add advertisements, but that also breaks the secure connection that Messenger tries to use (man-in-the-middle attack), and hence Messenger will refuse to operate.
If I am right, then some possible workarounds to make Messenger bypass this service:
Explicitly use a different DNS server, like OpenDNS or Google DNS.
Add the domains that Yahoo! needs to your hosts file, to map them to Yahoo!'s IP address directly. (But: how to know what domains it needs? And what if they change? Better try a different DNS server.)
Both workarounds might not work at all, if the VPN server you use does not allow for direct access to internet IP addresses. (And I guess they'll indeed disallow that, as relaying your requests is the only way they can add the advertisements.)
Can you provide any details on the exact VPN service you're using? Did you try other secure sites, such as https://gmail.com or some internet banking? And above all: can you access https://login.yahoo.com like suggested in your screen capture, without any security warnings? (You can try another browser for a first test as well.)
If you're using
thefreevpn.com (somehow, I don't really trust that site) then here's a note from the main developer, which makes me think that HTTPS over its default port should be no problem:
currently we allow SSH, FTP, HTTP/HTTPS, ICQ.