The Healthy Minds Program works with Apple Health and logs "Mindful Minutes". There are a few steps to set this up.
First, you need to give permission to Apple to record data from the Healthy Minds Program App. To do this, go to Settings > Health Data > Data Access & Devices > Healthy Minds > toggle Mindful Minutes ON under "Allow Healthy Minds" To Write.
Next, open the Healthy Minds Program app and go to Profile > Settings (gear icon in the upper, right-hand corner) > Apple Health > toggle ON.
Now that permission has been granted, Apple Health logs any time spent listening to Practice activities in the app. Time spent listening to Lessons (Learn activities) is not counted towards Mindful Minutes. To see your data, go to Settings > Health Data > Data Access & Devices > Healthy Minds > Data from "Healthy Minds" > Mindful Minutes.