An extensive review study of the scientific literature has concluded that swallowing capsules of evening primose oil and borage oil will not relieve your eczema symptoms.
Evening primrose oil and borage oil are "gamma-linoleic acids," or long-chain fatty acids that have a kink in the sixth bond in the chain. They are also called omega-6 fatty acids and have long been touted as beneficial in reducing inflammation and treating autoimmune diseases when taken in capsule form.
A team led by Joel Bamford at the University of Minnesota Medical School
read through 27 academic studies, in which 1596 patients had taken
part, and concluded that taking evening primrose oil or borage oil had
provided no benefit.
I've taken evening primrose oil capsules myself. They were expensive and did nothing for me.
This review seems like it could be the final word on the issue. However, the overall total number of participants is quite low--each study would have had around 50 participants. So it's possible that the authors of this review have just combined a whole lot of garbage results and the overall conclusion is not solid...but if there were some kind of real beneficial effect you'd think a review of this size would detect it.
I'm skeptical of supplements in general. In my experience, eczema is a condition in which diet is indeed important--but the importance lies in leaving out foods that can trigger allergy or inflammation, rather than in consuming certain foods and expecting them to make your skin better.