In all of the MIME::Lite code that particular error message only appears once, as shown. Net::SMTP->supports('AUTH'... fails, that error is generated. So, why does it fail from the server when it succeeds on your workstation. Same Perl, same modules; time to look elsewhere. Is the server blocked from connecting to gmail.com? Since this is email, is something on your network redirecting email ports to local email servers?
On the server, at a DOS prompt, try:
> telnet smtp.gmail.com 25
If you get a connection, you'll see something like 220 smtp.gmail.com.... If you can't connect, or get a local email server - there's your problem.