Yes, for sending. No, for receiving.
Only one email server can receive messages for a given domain name. It could be either Mailgun or Google servers, but not both.
However, you can use the same domain for sending at multiple servers. If you’d like to register your Domain at multiple servers for sending but you don’t want to receive email at Mailgun, just don’t configure your MX records to point to Mailgun.
If you are receiving emails elsewhere with your domain, we recommend using a subdomain at Mailgun so you can also receive emails at Mailgun. This helps improve deliverability and allows us to more easily deal with any issues that arise with recipient email servers.
Without these records you may see an increase in "Sender Domain Verification" errors, which are errors that the recipient server returns whenever your domain lacks MX records. By configuring your domain with MX records (Either Mailgun's or your other incoming server) it will resolve the above error and prevent it from occurring.