Hallux rigidus is a painful stiffness of the big toe joint of the foot. The best shoes for hallux rigidus are going to be those shoes that do two things:
1. Restrict the movement of the joint so that it is not so painful to walk
2. Facilitate the forward movement which is harder if that joint does not move too much
This means the stiffer or more rigid the shoe is across the ball of the foot, the less that that big toe joint is going to move, so look for a shoe that is more rigid. Shoe that have a rocker in them also restricts the motion of the joint and helps facilitate forward movement. They do this as you can rock across the shoe to move forward rather than bend that painful big toe joint. Some shoes have rocker action that is built into them, such as the Hoka One One running shoes. Other shoes can have rockers added to them on the outside of the sole.