Yes, in particular it just does a HTTP GET request which pushingbox.com can interpret as a command to post an event. It is just like when you go to a web page in your browser.
That is why I asked if you had to do anything special for the Twitter example--it uses a service similar to pushingbox.com but for Twitter. Do you use a proxy to connect to the internet for example?
My code completely throws away the response from the web server (pushingbox.com in this case) and assumes it has connected. You should probably look at client.available() and client.read() the response from the server and print it to the Serial port (over USB) to help debug this problem.
In your example from gmail, it says:
This means that gmail requires an encrypted connection using TLS via the STARTTLS command. The Spark core cannot directly send email using gmail service since it cannot do TLS encryption right now. You must use a service on the internet to act between the core and gmail. Pushingbox.com is one such service, but Spark will be introducing their own service that can do this in the future.