How can I prevent a forced iOS7 upgrade when restoring iPhone 4s from backup
You may be stuck on iOS 7 if you don't have a way to get that software from Apple's servers. Typically only registered developers can download several versions of software, but if you search - there are many places that archive these software ipsw files. It's always a bit of a risk downloading software from a place other than the vendor, but I don't know of a better option.

Typically, when a new OS is released, iTunes 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 iOS 6 and still has it on the hard drive? 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 iOS 6 backup on iOS 7. Also - this is more about the computer and iTunes having the old iOS file already downloaded than something that's inherent on the device. Any change you have a backup copy of the iOS files iTunes downloaded?