Must I enter the Schengen area using the E-gates?

I believe it is discouraged to avoid them if you are eligible, but there is nothing in the Schengen Borders Code that mentions automated passport control one way or the other. Practice may vary from one country to another, since electronic gates are implemented by each country rather than by a central authority.

Last summer, I was leaving the Schengen area via Amsterdam Schiphol with someone who had a non-biometric EU passport. We went to the desk, but approached separately. As the officer handed my passport back to me, he asked if I was traveling alone. I said "no" and that was the end of it. I suspect that if I had said "yes" he would have suggested that I use the gates the next time. But since this happened after he processed me, he obviously wasn't going to send me to the gates because I lacked an adequate reason not to use them.

To echo Mark Mayo's answer, here's a quote from Schiphol's website, with emphasis added:

Cross the border faster with eGate

eGate is an automatic self-service border checkpoint. You cross the border faster than with a manual inspection because you scan your own passport. You can use the automatic gates if you are 16 or over and hold an EU passport that contains a microchip. Follow the instructions at the gate for a smooth passage. Watch the video below to see how the system works.

Similarly, from the German federal police (I found this link on a page from Frankfurt airport):

What is EasyPASS?

...

This free of charge and voluntary service is an important component of integrated border management in Europe. The EU supports the new control system with funds from the European External Borders Fund.