I hear all these horror stories of bodies rejecting implants and patients getting sick. I have eczema. Will this be an issue?

There is no scientific evidence that implants cause or are associated with any auto-immune disease.  However, there are a number of patients who describe various symptoms that they put own to their implants.  I have never seen a patient where eczema is problematic when they have their augmentation.  Discuss your concerns with a Specialist Plastic Surgeon.  All the best!