Cryptocurrency serving as hash breaking? There are many people mining for Bitcoin right this second.[Citation needed]

Bitcoin's proof-of-work mining system is based on a hash function (SHA-256, I believe), and there are ASICs built specifically for mining this. 

Would it be possible to engineer a cryptocurrency such that the calculations of the mining process could be used to break hashed passwords? Without the miners knowing?

And, assuming a worldwide mining community approximately equivalent to Bitcoin's and with dedicated hardware for the purpose, would this be a feasible way of breaking hashed passwords?
Maybe

First let's say what you got correct:


ASIC is indeed used to mine, and mining is more or less calculating SHA-256 over and over (password hashing schemes use some things to make it somewhat harder for ASICs, but this won't be problem here - miners will use whats best).
If large-enough mining community formed, breakage of passwords could be eased.


However, there would be some problems that could prevent acceptability of such cryptocurrency:


It won't be possible to make it really without knowledge of miners, or at least some suspicions: To claim that you found block X, you would need to publish what password it was. It would be highly suspicious that some solution would be "Password123" "admin1" etc.
You would need database of passwords to break. Normally, this is done by some kind of mathematical rule. You instead would publish a large database and claim it is random. This again would be suspicious. And if you allowed yourself to add more hashes to database, then it would be suspicious, as you had a way to easily mine any amount of this cryptocurrency for yourself (you can generate password, upload it and then immediately solve it yourself - you can however put that aside when you pay for blocks).


So in the end, it doesn't look that easy, but if people somehow didn't realize these red flags, it should be fairly easy to do (not including trust problems - only one cryptocurrency is widely-accepted so far - bitcoin).
Also, if some password was really strong, whole bitcoin mining processing power won't help you - it just will be far too hard.