If you have shell access to the server, or if you are comfortable testing things in PHP scripts, you could try a direct connection from the server to smtp.gmail.com to see where the breakdown is. I would check:
- Does smtp.gmail.com resolve correctly? (Currently I get a CNAME of gmail-smtp-msa.l.google.com with addresses of : 2607:f8b0:400d:c0e::6d, 74.125.29.108, and 74.125.29.109)
- Can you get to just gmail.com on port 80? (Different host, but confirms external non-proxied access)
- Can you get to smtp.gmail.com on port 587 via a low level tool like bash or telnet? (You don’t expect a valid SMTP request/response, just something other than a network error.)
- Can you connect via SMTP to smtp.gmail.com:587 with a tool other than PHP/PHPMailer? (Maybe PHP itself is blocking the connection? Unlikely.)
Ultimately, this sounds like a server/host level error, and you will probably need to ask your system administrator or hosting support for resolution, but any of the above questions/answers will help them identify and correct the problem.