Not all Illustrator features currently translate into vector artwork in Character Animator.
For Illustrator artwork, you can use the "Render As Image" checkbox in the Properties panel to have it render exactly as Illustrator, but you'll lose the vector features (like the Wiggler).
* Double-click the puppet to open it in the Puppet panel
* Select any artwork layer (gray box) in the Puppet panel outline view
* Check the "Render as Image" checkbox in the Properties panel.
More details here:
Alternatively you can expand the appearance of your Illustrator artwork before using it in CA. I prefer to do this as it keeps it all vector based. If you have a pattern that cannot be expanded a workaround is to use Illustrator's Image Trace feature to "vectorize" it.
One additional note, shapes with gradients that are rasterized in Illustrator also seem to look a bit odd in CA (in case anyone were to attempt that workaround). The "Render as Image" checkbox suggested above seems to work best for me for gradients.