In-app purchases made in the normal versions of the Angry Birds games are not transferable to the HD versions and vice versa. Unlockable in-app purchases, such as the Mighty Eagle are transferable between iOS devices that have the exact same version of the game installed. The exceptions are all consumable in-app purchases, such as power-ups, which are not transferable.
When you download the Mighty Eagle or other unlockable item onto an additional iOS device, you will not be charged again. First you will be asked if you want to purchase the item. Tap "Yes", and then you will be given the price etc. But once you type in your password in the iTunes Store you will be notified that you have already purchased the Mighty Eagle for that particular game and you will be asked if you wish to download it again. Tap "Yes", and you will get the feature for the additional iOS device! Of course you must be using the same iTunes Store account (Apple ID) that you originally used when purchasing the item to get it back.