Hi Eddie,
first, if you're using Moodle 2.5.+ please install a nice mail tester tool: https://moodle.org/plugins/local_mailtest .
The tool will help in troubleshooting any communication issue between Moodle and the SMTP server.
That being said, if I look at https://help.mailgun.com/hc/en-us/articles/203380100-Where-can-I-find-my-API-key-and-SMTP-credentials- your settings look fine so it could be a network issue or... a cron issue i.e. your email are not delivered since they are not actually queued.
HTH,
Matteo