Hello richr604!
Welcome to the Apple Support Communities. Thanks for bringing your question about the Health app working on an iPhone 5C. Health is such a cool, multifaceted app so it's my pleasure to help you out today.
You don't mention specifically what the app is doing, or whether you can find the app on the iPhone 5C, so for now I'll provide general information and troubleshooting.
From the,
iPhone User Guide
Health, Get Started section.
Health automatically counts your steps and walking and running distances (not available on iPhone 5). To gather additional health and fitness data, you can use other apps (such as nutrition and fitness apps) and devices (such as blood pressure monitors, weight scales, and glucose monitors) that are compatible with Health.
Install third-party apps. To see recommended apps, tap a category, such as Activity. Tap a recommended app to download it from the App Store. After you install and set up the app, you can choose to share data with Health.
If an app you installed unexpectedly quits, stops responding, or won’t open
Please note that Health is not an app that can be deleted or reinstalled.
Use the Health app on your iPhone or iPod touch
Take care!