The type of kick I was referring to is the flutter kick, not the whip kick. Sorry
This is an outstanding question. Many great swimmers have had problems with their flutter kick. You are not alone. I have known some sprinters who actually go backwards as well when they kick with a board. However, they were top swimmers in the nation with the whole thing. Don't worry so much about your kick by itself.
The flutter kick is an extension from your hips. It is like a whip (as your referred to it as the whipkick), however I like to call it a wave. your legs needs to become like a wave going up and down. It starts in your hips...then through your knees...then to your ankles...and out your toes. The key is that you don't just kick in one direction. You need to kick up AND down.
It's very hard to critique someone without looking at them. I do have a drill that could help you out to see what progress you make. Try holding the wall while kicking. Just kick while holding the wall. Try and make little changes in your kick. You will feel if you make any forward motion by the force you are putting on the wall. I hope that makes sense. It's just easier to do this rather than kicking with a board. If you start "going backwards" you will feel yourself separating from the wall. You want to prevent that.
Another thing you may think about is your ankle flexibility. Much of the problem with backwards kicking is the inability to relax your ankles. You really need to make sure they are relaxed and as pointed as they can be. This may be tougher for you. Remember though...kicking doesn't dictate how good of a swimmer you are.
I hope that this helped you out a bit. Please let me know if you have further questions.