I was first who did it ;)
Part 1 (editing: Sony vegas + After Effects): 0:20
Part 2 (Comressing): 5:18
Part 3 (Uploading): 7:13
SUPER Download Link: http://www.filehippo.com/download_super/
Instagram Profile Link: http://instagram.com/dmitriysa
If it is not working, on the last repeat step 3 again - 100% will work.
It's took a lot of time to make this tutorial, but finally it's ready to watch. If you have any questions, leave comments below.
Sorry for a little bit longer video. Hail from Russia!
[Update]: Some time ago Instagram finally accepted to load the video from the gallery to your account. Now if you want to make the process a little bit easier, on the last step you need to put your movie to the gallery of your device. The path is: var/mobile/Media/DCIM/XXXAPPLE and then load it from the app on your phone or tablet. If you will follow this way you can make it 720p (or already square size - 640x640) - Instagram can crop the video to the required size.