For now, yes. Apple will likely release an adapter for third-party headphones at some point (just as it's promised to do with the remote-enabled third-generation Shuffle headphones), but there's no telling when that might be. For now, you're required to use the Apple earphones because they're the only earphones that have the hardware built into them to activate Voice Control using clicks on the remote attached to the cord.
Luckily for those who prefer to use headphones other than Apple's earbuds, there is another way to activate Voice Control: the home button.

