Yes! You can use FeedBurner feed services with AWeber's Blog Broadcast. Keep in mind that Google no longer offer support for any FeedBurner RSS feeds.
Using FeedBurner feed services, the Blog Broadcast identifies new posts based on their URL. Be careful not to make any changes in FeedBurner that cause them to republish your feed, or to change the URLs for your posts. This could cause the Blog Broadcast to think those posts are new and need to be emailed.
Switching From a Non-FeedBurner Feed to a FeedBurner One?
It is a good idea to manually review your Blog Broadcasts, at least at first. When you change your feed URL, Blog Broadcast will think it's an entirely new feed, with new posts, that all need to be emailed.
Manually review your feed until Blog Broadcast has read your previous posts and created Blog Broadcast drafts (which you can just delete). After setting up with your FeedBurner feed, feel free to switch your preferences to send automatically.