When the bitcoin fork happens, is something like this likely to happen as well? Could BCC similarly appreciate in price?
I don't have a crystal ball, but shouldn't everybody and his uncle be making sure they get whatever BCC is due them?
Sure, BCC might appreciate in price, like any other alt coin. The thing is, with ETH/ETC, the overall market cap dropped significantly--something like 40%. Any significant network split should in theory cause a loss of value for both networks. It's really difficult to predict future demand for Bitcoin Cash, but it seems that exchanges have indeed received interest from their customers.
Just make sure you're careful about splitting your coins. Last I heard a couple days ago, there was no mandatory replay protection in BCC clients. I'm sure many bitcoin holders will try to access their BCC, and will lose bitcoins in the process of moving them.