First, you should be using the Adobe Media Encoder to render your H.264 files because you can get better compression, higher quality, and multi-pass rendering.
If you really want to use the render cue then you need to edit the output module and set up a custom H.264 preset then make that the default. Here's mine. I have set the default to a Tiff sequence. Find the option under the Edit menu:
I tried your suggestion of using a Output Module Template and it worked! But this brings up a second question: Why does the Media Encoder have a better H264 output? Don't both H264's put out mp4 files? Or are they different somehow? I don't really understand how to set up Media Encoder for rendering. Is there a tutorial somewhere? Thanks!
The Adobe Media Encoder uses better under the hood software and allows Multi Pass rendering which is critical for good Intra frame compression. Because of the way the output module serves up frames it is impossible for the Render Cue and the Output module to do multi pass encoding. You'll just get much better results and as a side benefit, you can load the comp in the AME and then move on to another project while you are rendering. That means no wasted time. I hardly ever wait around for a render. I work on creating content more than 90% of the time that I'm in AE. It's must way more efficient.