Miners can't be repurposed to do anything except sha256d hash. It can't even do just the normal sha256, it must do two rounds of it.
It could possibly be repurposed to crack sha256d hashes, but there aren't many things (besides bitcoin) that use it.
Well... like I said there is a SHA-256 hash with a hint just waiting to be cracked, so this would be perfect if I could get it humming along.
Can I force-feed it the hash and make it think it's a network block?
Sha256 is different from sha256d. It is physically hard coded into the circuits to only do sha256d, not sha256. The difference is that sha256d is two rounds of sha256. So, you can't crack a sha256 hash with a miner since it is not sha256d.
You could force feed the miner with the proper data, it doesn't need to think it is a network block. But, it will only output the sha256d hash of the data, not the sha256.