To export an After Effects file as a playable quicktime, you need to render it out through the render queue.
The best luck I've had, in terms of file quality, was to render out a lossless QT file from After Effects, then re-export that QT file through MPEG streamclip with the proper video and audio compression settings that we need for projecting at the planetarium.
To render your file from AE, go to:
Composition: add to render queue
That will open up the render queue where your timeline usually is.
From there, choose:
Render settings: best settings
Output module: click on lossless, make sure audio is checked in the dialog box that pops up, and choose lossless for that as well.
Then, click on the "render" button at the upper right hand corner of the render queue.
Once you've rendered that QT movie, open that file in MPEG Streamclip, go:
File > Export to Quicktime
With video compression settings set to H264
And audio compression settings set to MPEG-4 AAC
Click on "De-Interlace Video."