Flaxseed oil has an inhibiting effect on some bacteria such as helicobacter. I think that I was drawn to it for that reason. You say that you have stopped eggs and nuts which are high protein and require stomach acid for digestion. Have you tested your stomach acid function? Have you had a test for helicobacter pylori? Helicobacter can cause Leaky Gut and it is not alone in doing that.
Avoiding foods because you think that they cause Leaky Gut is not helping you because they are not causing Leaky Gut IMO.
I don't think that flaxseed oil causes Leaky Gut either but I doubt that it will heal Leaky Gut. If you have a reliable source for it and it is fresh and you think that it is benefiting you then continue to take it.
Your body will be absorbing undigested food molecules which your body will be producing anti bodies to. They are then affecting your thyroid gland where your immune system regulation is weak. If you have hashimotos then make sure that you are getting lots of selenium, well above the recommended dosage. Taking high dose B12 might also help as it up regulates T suppressor cells. It works well for people who are primarily IgE anti body responders. Taking Methycobalamin works well for me and stops most autoimmune effects. It takes high doses of 20 to 30mg day