I suspect I have Haglund's Syndrome. There is a bump on the back of both my heels. It's almost as if they curve outward. I've had them ever since I can remember. They become painful when I wear shoes for a long period of time. Can they be removed?

Yes. They can be surgically removed and this is what I would recommend to you. Given your relatively young age you would appreciate many years relief. Now having said this, there are complications still to consider. This should be discussed with your doctor. Conservative care would work but you would be involved with this the rest of your life and you would be limited on your choice of shoes. .