Bitshares and EOS are different platforms. EOS is designed to be a generalized smart contracts platform like Ethereum, but borrowing some of the concepts from Bitshares and Steem (both were founded by Dan Larimer who is building EOS).
So, in answer to your questions; If you want to be part of EOS, you need to hold EOS tokens. Holding Bitshares does not entitle you to any share of EOS. EOS is using the Ethereum platform to distribute tokens, so you can buy some Ethereum and then send them to a special address and this will reserve you some EOS tokens. See
https://eos.io for details. EOS tokens *may* at some point in the future also be available on the Bitshaes DEX (Distributed EXchange), probably via a gateway like OpenLedger.io (
https://openledger.io)
Second question; Although EOS was built by the same guy that built Bitshares, its not clear (yet) what concepts (if any) will be applicable to Bitshares. There are lots of discussions going on in Bitshares development circles about how Bitshares and EOS may interact (or Bitshares may enhance itself with new technologies being built for EOS), but since EOS hasn't actually been built yet it's a bit early to know. Certainly, in the past Bitshares has benefited from developments on other platforms like Steem (Not Steam) for example, so time will tell...