The good news is that you can dis-allow indexing, and is definitely something you'll want to do (and make sure your kids to the same).
Here's how to stop Facebook from indexing your information:
- Log into your Facebook account.
- Go to your SETTINGS and select PRIVACY SETTINGS
- Then click the SEARCH link
- Uncheck the box that reads, "Allow Indexing".
That should do it. However, while you're looking around your settings, you might also like to check some of the other settings as well to make sure nothing is being shared that you don't want.
For more information about Facebook Privacy, click here.