Well, clips and straps won't secure your feet as well as eggbeaters (I just switched in the other direction), but they can come close if you tighten the straps down hard. Then again, that means you might have trouble getting your foot out...

As long as you realize you can't exert pressure straight backwards (that's how you pull your foot out!) you shouldn't have any problems with your feet slipping out. This means you only get power into the bike for ~3/4 of a rotation, but you're only missing the bits when your cranks are vertical...still plenty to stop quick, skid, whatever.