No. The math involved in cracking a private key uses a completely different set of mathematics and source data. There is no benefit in reusing one for the other.
There is a conspiracy theory going around about the SHA hashes being a big rainbow table for "The Russians" but that crackpot theory has no basis in mathematics or applied usage. I hope people keep talking about it so I can buy BTC while they are still cheap. FUD (Fear, uncertainty, doubt) are good to keep BTC's price low while before the market value becomes what it should be.
To be more specific, SHA hashing (mining) is a different set of math used for the ECDSA secp256k1 encryption that in the private key.
Now that ASICs are used, it's impossible to dynamically re-wire the hardware for such a purpose. In theory FPGA can be reprogrammed for this, and so can GPU/CPUs.
You could keep all your BTC in one address, or you could diversify your risk by keeping BTC in many different accounts (addresses) knowing full well that this will increase your transaction size and therefore your transaction fee. The choice (to some extent) is yours