This will erase everything off you device and return it to the factory settings.
It's essential to back up your i Phone and/or i Pad every time you update the device's software. Follow this guide on backing up, restoring, and updating your phone and tablet.
The i Phone 5 and later, i Pad (fourth generation and later), and i Pod Touch (sixth generation) can run i OS 10. Plug it into your computer, open i Tunes, and the model as well as the currently installed version of i OS are listed under the Summary tab which has a phone icon in the top left.
If so, you can decline the offer to upgrade and restore the device from that version of firmware.
As was covered in the comments - there is no restriction to load i OS 6 backup on i OS 7.
As bmike also said, you should be able to restore a backup made on i OS 6 to i OS 7, so that makes the entire problem moot anyways. Apple does not sign any i OS 6 version for that device anymore.
ANY i OS 6 and 7 version before 7.1.1 does have SERIOUS security flaws.
Typically, when a new OS is released, i Tunes will only download the latest software if you don't already have an older version of the software stored locally.
Is there any chance that computer (or another computer) downloaded i OS 6 and still has it on the hard drive?
You will have to download firmware and then attach your phone to your computer.