This single requirement will stop me from trying their app. I have never, nor will I ever, use my FB login for another companies service. If that's the only way to log in, I won't bother using the service.

A fair approach would be to allow me to use the thing for a day without "liking" it. Then they could say "Like it so far? Then click the Like button and listen on!" and only after that keep me out if I don't.

I don't think they'd get much fewer likes this way and at least they would be genuine. Plus, since more users would sign up to test the service, perhaps the numbers would even play to their advantage.

They have ~1.3 million people (at the time of this comment) who have liked it and joined the beta. The privacy concerned outliers are likely much lower in number than the additional people gained by the viral FB network launch. How many other services have that type of adoption on a first day beta?

Isn't it against Facebook policy to use the like button in this manner?

http://developers.facebook.com/policy/

You must not incentivize users to use (or gate content behind the use of) Facebook social channels, or imply that an incentive is directly tied to the use of our channels.