I would recommend Exodus. I don't know if you'll need to give out your 12 word seed phrase to claim your new EOS Tokens after the swap...I've not heard that. Even if you did, you could delete the Exodus wallet completely after the swap and download a new one with a new seed phrase. Just make sure you obviously move your coins off it before you do.
There's 2 reasons I'd recommend Exodus. The first is so you can claim upcoming airdrops. I think there's 2 or 3 more before the EOS mainnet launch. The second reason is because Exodus is obviously safer than keeping it on an exchange. That said, Binance is one of the safest exchanges, so you should be ok either way.

As far as tax implications, I don't know where you live, but I'd be surprised if you were taxed just for moving coins from one exchange to another. That should be the same as moving it from one wallet to another. It shouldn't trigger a tax event unless you sell.