gymreading on "[Plugin: WP Mail SMTP] Tried "Send Test" -- Nothing happened!"

I'm trying to use this plug-in with Gmail. I click "Send Test" and it looks like a new page is loading (for a long while) then I get:

Test Message Sent
The result was:
bool(false)
The full debugging output is shown below:  (lots of gibberish...

Final lines of the gibberish are:

2016-04-09 19:40:17	SMTP ERROR: Failed to connect to server: Network is unreachable (101)
2016-04-09 19:40:17	SMTP connect() failed. https://github.com/PHPMailer/PHPMailer/wiki/Troubleshooting

Can someone tell me what I'm doing wrong? (In plain language please. I'm no techie.) I'm pretty sure I'm following the instructions in the faq exactly.

For what it's worth, I also have Contact Form 7 on my site, and ultimately I hope to use this plugin so my forms send messages encrypted via Gmail, instead of unencrypted via mail(). Am I barking up the wrong tree?

Here's what I've entered on the "Advanced Email Options" page:

From Email: mygmail@gmail.com
From Name: MyName
Mailer: Send all WordPress emails via SMTP.
SMTP Host: smtp.gmail.com
SMTP Port: 465
Encryption: Use SSL encryption.
Authentication: Yes: Use SMTP Authentication.
Username: mygmail@gmail.com
Password: mypassword