What maladaptive behaviours or cognitive distortions develop as a result of chronic childhood illness?
There are many psychological models for abnormal behavior so there are many ways to answer your question.
The Stress-Vulnerability model of mental illness is a well established biological theory. It states that biology provides from vulnerabilities to mental illnesses which are activated by stress and trauma. It would dictate that the stress/anxiety caused in the past and present by a bad childhood could result in any mental illness based on what biological vulnerabilities were present in the person. 

I think you should read "Chronic childhood disease: An introduction to psychological theory and research", "The occurrence of maladaptive health-related behaviors and teacher-rated conduct problems in children of chronic low back pain patients" and "Abnormal illness behaviour: childhood experiences and long-term consequences", "Relating Parent and Family Functioning to the Psychological Adjustment of Children with Chronic Health Conditions: What Have We Learned? What Do We Need To Know?"