by "eating disorder cognitions", one is normally referring to the cognitive aspects of eating disorder psychopathology. They can refer to fear of gaining weight, body image disturbances (viewing oneself as fat even though one is underweight), undue importance of body weight and shape on self evaluation, body dissatisfaction etc. For instance, in the eating disorder examination questionnaire, two subscales (weight and shape concern) deal specifically with some of these cognitions.
It refers to cognitive aspects, in eating disorders there are specific cognitive features, usually, high value of shape and weight in self evaluation, concerns about weight and shape, the internalization of a thin ideal, feelings of guilt after eating, sense of lost of control when binge eating, body dissatisfaction.