Yes! You can use FeedBurner feeds with AWeber's Blog Broadcaster.
By doing so, you'll be able to boast higher subscriber numbers in your FeedBurner chicklets — we report the number of email subscribers you have to FeedBurner, and they aggregate those with your RSS subscribers! The Blog Broadcaster 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, as this could cause Blog Broadcaster to think those posts are new, and need to be emailed!
Switching From a Non-Feedburner Feed to a Feedburner One?
In that case, it's definitely a good idea to manually review your Blog Broadcasts, at least at first, for the same reason as above — when you change your feed URL, Blog Broadcaster will think it's an entirely new feed, with new posts, that all need to be emailed.
Manually review your feed until Blog Broadcaster has read your previous posts and created emails for them (which you can just delete). After that, feel free to switch your preferences to send automatically.