So I know you can store hashed information on the Ethereum blockchain.
Lets say I forked blackcoin a POS version of bitcoin (https://github.com/CoinBlack/blackcoin). Can I also store hashed information on the POS blockchain?
Consensus method (Proof of Work vs Proof of Stake) does not impact the functionality and features of the blockchain.
However, Bitcoin was not really intended to store information other than transactional data really. So while you technically you could store data on a blockchain based on bitcoin (for example by hashing your data into an address and sending a small amount of bitcoin to that address see here), there are better ways to do that.
Some protocols are designed specifically to allow for data storage and access. For example, creating a simple data storage smart contract in Ethereum, or even simpler, explicitly sending data in an Ethereum transaction.