They probably backed up the IE favorites, Firefox stores its bookmarks elsewhere. The bookmarks are stored in a file called places.sqlite inside the profile folder. Did the techs backup the profile folder? For details of its location see http://kb.mozillazine.org/Profile_folder_-_Firefox

If they did backup the profile folder, you can copy places.sqlite from the backup and place it in your current profile folder.

I tried to click on the places.sqlite file but it asked me for a program to open it with. It then took me to a sqlite web page. I don't know if the techs backed up this file or not. All my internet explorer favorites were there but I guess they get saved to a different place. So I don't know if I can copy that file since my computer didn't know what program to use to open the file or if I did something wrong. Any suggestions? Also, how do I find the current profile folder and put the copied file in that one.

Where did you find the places.sqlite file that you tried to open, if it was not the backup then you are probably looking in your current profile folder. Another link that shows how to find the profile folder is https://support.mozilla.com/en-US/kb/Profiles

You do not need to open the file. If the techs backed up that file, you just need to copy it into your current profile folder. If they did not backup your old profile folder you will have lost the bookmarks.

You can find the curent Firefox Profile Folder via:

Help > Troubleshooting Information > Profile Directory > Open Containing folder

Did the techs restore all files and folders that were backed up or do you have the backup in a file or separate drive?

Where (which location) did you find that places.sqlite file?