1. Less is More
The best case scenario is one where you walk in to the receptionists office and someone greets you by name and confirms your appointment without any input from you.
This would be very hard to accomplish with so many offices over-populated and under-staffed. The good news is that most people realize the situation and are comfortable with using technology as long as it isn't frustrating.
I have seen way too many forms in doctor's offices that look like they were written for medical professionals. Do you really need to know if anyone I'm related to has glaucoma? I barely know what that is.
Look for an option that only requires a single piece of information to check-in.
2. Be professional (not quirky)
Healthcare is something every single individual has to deal with at some point in their life. Some more than others. It can be a sensitive issue so be sure the interaction with the user is unassuming and professional.
Chances are good that most users of this device don't want to be there in the first place so anything you can do to not frustrate the user will go a long way.
3. Reassure the user / Leave the door open
The welcome screen should have a short message letting the user know their data is kept safe and only authorized personnel will be able to see it.
I would also put a message like this, "Feel free to give this device back to a receptionist and let them know if you prefer checking in another way. We understand."
4. Recall over Recognition
Pretty much everyone knows their name so I would start with that. A single input box asking the user to enter their name (please no auto-complete or help here) In the off chance that 2 people with the exact same name have an appointment around the same time you may have to do something extra but only in this extreme case should you ask for more input.
Once the system knows the user it can confirm their appointment time. You should know this so don't make me enter the exact time -- I gave you my name, just let me confirm my appointment.
5. Be careful with medical data
If you can avoid confirming personal information at all then that would be best. In the case where you do need each person to confirm / update personal information then I would say a receptionist would need to log each person in to the system where the user can then see all their previous answers and confirm with a single click that they have reviewed the information and it is accurate.