First, I'm not sure which server GogTasks is trying to reach when it fails. GogTasks communicates both with the Google back-end and the services of GogTasks (read more about it in the
privacy policies).
Being able to reach GMail through a normal browser is an indication that you can reach the Google services, but you should also try the other URL (1) I list below. If you additionally try the GogTasks service (2) through the browser, we'll be able to see if the proxy stops you or if this is a problem with the GogTasks configuration or the app itself:
First you can try to enter the settings you have in Internet Explorer manually in the GogTasks connection settings pane. This should normally not be neccessary, as it is exactly what "Use settings from IE" is supposed to do. I have had experiences that it does not work for all my customers though. So if you could start Internet Explorer, go to Tools > Internet Options , Connections > Lan settings and copy the proxy server settings over to the connections pane of GogTasks - see if that helps. See attached image proxy-settings.png for illustration.
Second you may try to enable "use proxied license server" under the Advanced tab of the GogTasks settings.
Please let me know if any of this yields any result.