QuickTime H.264 is no longer being worked on or supported by Apple and it's a lousy container. You should not be using it. There are many known bugs especially with certain multicore processors.
The better choice is to render an h.264 MP4 using the Adobe media encoder.
Yes, but as I said, the option is grayed out for me. It won't allow me to send my composition to Adobe Media Encoder
I see file name is something like "Shot 123 FINAL". That makes me think you'll use it in another application... like an editing app.
Do yourself a favor and forget about h.264 -- which is VERY lossy -- and go with a QT in ProRes 422, Photo JPEG, PNG or Animation codecs. You won't lose image quality.