Will I quit WoW permanently over this change? No. But I will also never again participate on any Blizzard forum. Scott over at Broken Toys sums up my feelings pretty much, and presents a reasoned defense of online anonymity.
The folks running around calling people who have a problem with this change 'irrational' and 'paranoid' must be really happy in their privileged denial bubbles where the world is always safe and where losing your privacy online couldn't possibly have any negative repercussions.
Plus there's the fact that MMORPGs were originally based around creating an 'alternate persona' separate from real life, so why would folks be surprised that a lot of people who enjoy that aspect would consider bringing RL into their game as a sort of violation?
EDIT: Oh look, Blizz made a deal with Facebook to share RealID info across platforms. So at least now we know the monetary reason for this change -- the claim that it's all 'to improve the forums' is just PR bunk. You don't need to strip away pseudonyms in order to properly moderate an online community.
EDIT2: After the enormous outcry provoked by this announcement, Blizzard has pulled the idea.