No they don't, since the two signatures they would make for different branches would get posted to the chain as evidence [1] and their stake (a minimum of 32 ETH) would get forfeited.
I don't understand. Say you have 32 ETH, and the chain is split. You now have 32 ETH-old and 32 ETH-new. You can use both these coins with no consequences, unless it isn't an actual split, but a sidechain.
In theory, if you validate on the chain that Vitalik doesn't like, he will destroy your 32ETH on the good-boy Vitalik chain.
You know, compared to ASICs, that allow you to go with whatever is currently more profitable and switch back at any time (freedom), if you don't behave like a good boy, Vitalik will steal your funds in PoS.