No, they are totally separate blockchains. To claim your BTX, you must go to the claiming service mentioned in the OP and follow the instructions. The snapshot was taken of the Bitcoin blockchain on block 463604, which happened on 04/26/2017. You must claim with an address that you can make a signature with that had Bitcoins at the moment of the snapshot.
i know they are separate blockchains, but don't they use same address format for pub/priv key?
Yes they do. However, sending BTC to your BTX address will not result in you getting BTX, nor vice versa. Even if you import the private keys of BTC wallet into BTX and vise versa. All that will happen is that you will spend fees moving coins to yourself. BTC coins and BTX coins have separate Coinbase hashes and separate transaction hashes associated with them.
Edit:Maybe I am misunderstanding what you want to accomplish. If you want to know if you can use the same keys for a paper wallet/cold storage. Yes you can, I believe. You may want to test a very trivial amount of BTX to verify. Just know that if you do that, if your cold storage device(s) becomes compromised, you risk both BTC and BTX. Also, your anonymity is somewhat less because if someone knows your public key for one, they know your public key for the other.
