Either will do just fine, especially if you intend to sync your framerate to the stream output but a few other points:
The 7700K is now getting old hat, and remains expensive, the later 6 core i5 8600K is probably a better bet.
If you game at other times without streaming on a fast HD display Intel still has a FPS advantage over AMD.
With any Intel 'K' CPU you will need to include a CPU cooler, the 2700X comes with an excellent boxed cooler.
Overall an AMD build will probably be a little cheaper.