I lost some files from my JVC Hard Drive Video Camera. Is there a way to restore them?

Yes, you do have ways to restore them back as long as you've not overwritten them with anything new.
Plunge this hard drive to your computer and download a hard drive data recovery freeware to help you:
http://www.freeware-fix.blogspot.com/2013/02/external-hard-disk-recovery.html
This hard drive data recovery freeware has offered four data recovery options for the user to cope with almost all data recovery problems, such as the deleted or formatted data recovery problem, the RAW hard drive data recovery problem and even logically corrupted drive data recovery problem. So, I believe you also could perfectly restore your lost files back from this drive with this freeware.
So, you should try it here:
http://freeware-fix.blogspot.com/2013/02/hard-drive-recovery.html

You have chance to get back some or all of those lost files if you take care.

First things first: DO NOT take any more videos or photos with the hard disk (on your camera) that you want to recover files from. Take the hard disk out of your cam and put it in a safe place. Because if you take any more videos or save any data to your hard disk, you may quite possibly ruin your chances of file recovery.

Next take this softwarecalled Wondershare Photo Recovery(also recover Videos and music files) to deal with the issue and see if you can find some of your lost files back. Get the corresponding versions here,
Windows version
Mac version

You should remember - save the recovered files to another disk . And before your files are recovered, DO NOT put more files to the drive as it is very important.

Yes, you can try some data recovery softwares such as Pandora Recovery, if the hard drive uses some other file systems, you can look for them, but they might not always be free.