I'm not sure about Firebase, but Azure is defiantly supported according to this page: https://www.mendix.com/platform/cloud/
Although with a free app it says you can follow this doc: https://docs.mendix.com/developerportal/deploy/private-cloud-deploy/#422-add-environment
to deploy to a private cloud node. This is mentioned in there: “Enter a Subscription Secret if you want your app to run as a licensed app. Without a license, your app will be subjected to restrictions very similar to those listed in the Free Apps section of Mendix Cloud.”
Also in the pricing, private cloud is not mentioned for free apps. https://www.mendix.com/pricing/
Not sure to what extent this is supported for free apps.