The criterion of libertarian paternalism requires that a policy do two things: 1. It must leave people free to choose their own actions; this is the libertarian part of the criterion. 2. It must guide people toward making a choice that a paternalistic observer would believe is good for them. Nudge policies generally do meet the criteria of libertarian paternalism, because it is only the choice architecture (the context in which choices are presented) that is being changed. No particular choice is mandated or prohibited; all individuals are free to choose as they see fit. Some (most perhaps) will be influenced by the nudge to make better decisions, and some will not be influenced by the nudge.
Behavioral economic policy is policy that is built on the building blocks of behavioral economics: purposeful behavior and enlightened self-interest. Behavioral economic policy takes into consideration predictably irrational behavior.
Firms have used nudges for a long time to alter their customers' behavior. Grocery stores strategically place their merchandise in such a way as to nudge their customers to spend more: they place milk at the back of the store, they place high-profit-margin goods at the checkouts, they put all organic foods in a separate section so customers won't compare prices to non-organic options, etc.