I found Imagick::clipPathImage and Imagic::clipImage did not work as I had expected. I thought they would just clip the path and throw away the extra data and you are done. Not the case. 

Here is how I was able to use a clipping path:

<?php
      $img = new Imagick("/Path/To/Test/Image.psd");
      $geometry = $img->getImageGeometry();

      // Uses the first path as the clipping path
      $img->clipPathImage("#1", false);

      // Fill the clipped part with a color
      $draw = new ImagickDraw();
      $draw->setFillColor("#000000");
      $draw->color(0,0, imagick::PAINT_RESET);
      $img->drawImage($draw);

      // Composite the clipped image with the old image. Set the color of the composite to any color you want to be the outside part.
      $composite = new Imagick($path);
      $composite->newImage( $geometry['width'], $geometry['height'], new ImagickPixel("white"), 'png');
      $composite->compositeImage($img, imagick::COMPOSITE_COPY, 0, 0);
?>

Then doing any resizing or creating thumbnails from the resulting image  disregarded all the previous commands so I "saved" it and started with a new Imagick object

<?php
      // Copy the image so clip is "saved"
      $clipped = new Imagick();
      $clipped->readImageBlob($composite->getImageBlob());
?>

I'm sure there is a simpler way, but this took me awhile to get right and there were some hurdles to cross so I hope it is able to help someone on the way.

This is all the convert equivalent of:

$ convert Test.psd -fill white -colorspace rgb -draw "color 0 0 reset" -clip -colorspace rgb -draw "Image Copy 0,0 0,0 'Test.psd'" OutputFile.png