You will use what's called an image link: <a href="http://pagelinkhere"><img src="http://imagelinkhere" alt="" title="" /> </a>
A typical feedburner code for email update would look like this: <a href="http://www.feedburner.com/fb/a/emailverifySubmit?feedId=yourfeedidnumber;loc=en_US">Subscribe to yourblogname</a>
You would edit that code and make it like this: href="http://www.feedburner.com/fb/a/emailverifySubmit?feedId=yourfeedidnumber;loc=en_US"><img src="http://imagelinkhere" alt="" title="" />
Replace the image link URL with the URL of the button you want to use.