There are two ways to find your BIOS information. One way is without downloading any software another way is download CPU-Z.
Both ways work fine.
Open Start -> Programs -> Accessories -> System Tools -> System Information. Here you will find System Summary on the left and its contents on the right. Find the BIOS Version option and your BIOS flash version displayed.
and you can find all information there.
What the technical support rep suggesting you to do is do a recovery of your system back to the state when your first purchase your eMachines. I believe you won't want to do this unless you don't have other choice.
Even if I download new BIOS (how I can tell it's new?)
If we know your current BIOS version and brand, we can easily go online and check if yours is the updated version.
Just a question, What's your Harddrive file system? Is it FAT32 or NTFS?
Mine is NTFS, NTFS is much better in terms of security wise. (did I get my facts wrong? lol)
If you are able to boot up normally, go to My Computer > Right click and select Properties. You will be able to see your File System being stated there.