No need to, it's just plain old cheap lawyer tactics, it's been around since the beginning of time:
Scare people (i.e. by making them feel they're in danger) and they will do what you want. (Law seems complicated to non-lawyers, so lawyers make use of that, basically by communicating "I'm the specialist and you don't know anything about this domain, so do as I say")
In the case of this story, all I'd suggest the owner of the app, to steer clear of legal issues, is:
a) don't use their name,
b) don't use their logo (or other very distinct parts of their Corporate Identity)
c) re-launch your app (using proxies)
EDIT: On second thought - they don't deserve your exclusive traffic/promotion/value you add. If you keep your app alive, you should multiply your sources (add as many similar sites as possible).