Have you got emergency contact details? Have you got a medical history? Have you got an iPhone?
Well then it makes sense to combine them.
That way if someone finds your iPhone lying around we have some chance of getting it back to you (the emergency details are available from the lock screen) and if we stumble on your listless frame, we have some hope of working out who you are if you have your phone on you, which you probably will...
This is also a good idea for you to encourage your students to do likewise, well those with iPhones anyway.
For more on the how, see the advice from the experts I've linked to below:
Health App—Medical IDWithin the health app, you can personalize the Medical ID to keep your important health information in case of emergency, from a lost phone emergency to a lost consciousness emergency.
Apple Insider guide.
Apple's more detailed breakdown on the whole app, not just the emergency contact element.