Shape from vector is not a 'perfect' conversion. My guess is that the issue is with the original illustrator file. Look at the 'U' shapes in the illustrator file and check they are not compound shapes or other 'complex' objects. If they are, simplify them into standard shapes and try again.
If that's not successful post another screen shot with one of the problem layers twirled down so we can see the contents of the shape layer 'contents' group and all it subgroups.
I agree with Mr. Abbott. There are limits to the conversion and presumably something is awry in your AI file in the first place. You have to provide soem screenshots of the AI paths selected and anchor points visible as well as the AI layers palette with teh relevant groups twirled open to understand their structure and the same for the AE shape layer hierarchy.
Nothing immediately jumps out at me as being odd - anyone else have any ideas?
A couple of further suggestions:
1. Just to try and eliminate the Ai file as an issue, save a copy of your Ai file as a much older version say 'Illustrator 8'. When the export dialog appears there's a menu to select the version at the top. Import that and see what happens.
2. If you still get the black screen twirl down the transform section for the shape layer and post an image of the Anchor Point, Position, Scale, Orientation, Rotation, and Opacity values.
Thanks Mike for the solution. I don't know why it worked to Save As an older file but it did. I tried the first solution from your couple of further suggestions.