Should I move ERC Tokens off of MyEther Wallet or can I store them there?
With a MEW you have full control over your private key. You can think of your private key as a mathemagical stamp that can be used to sign virtual "checks" written on the ERC20 contract's token.
ERC20 tokens are only stored in the contract that created them. The contract can generate virtual "checkbooks" for that token. The checks in the checkbook are not worth anything without a signature your private key can sign those checks.
Remember the ethereum network is only updated with signed transactions only private keys can sign transactions.
Also remember anyone who has your private key (or the list of 12-24 word seed) can sign transactions
Your private key has all the control. Your public key is basically like legal letter head, anyone can see it, but its also has no power without a signature.
My Ether Wallet does not have access to your account, it's just an interface for the network.