Utkatasana works your quad and buttocks muscles. With proper alignment you work the muscles of the outer thighs, hips, abdomen and upper body. You need to have the inner and outer thighs engaged equally to keep your knees right over your ankles. Do not let the knees splay out or in.
Eagle pose has been both indicated and contra indicated for knees in my research for this blog. I have been teaching Eagle Pose to my students with knee injuries and weak knees and so far they seem to be benefiting from this pose. It is a very challenging and energetic pose which teaches proper knee alignment which can protect against future injuries.
Other Recommended Poses
Revolved Side Angle Pose, Utthita Hasta Padangustasana, Parivrtta Utkatasanan, Supta Padanghustasana, Low Lunge (with back leg bent)
References: Yoga Journal