I have a history with depression and eating disorders. How likely is it that my children could develop these conditions?

Nature vs Nurture. To some degree, these are inherited - but i'm certain that they are also learned. I believe that part is most important. Firstly, because it is what we can effect. Life can be difficult too, so avoiding emotional/behavioral disorders isn't easy even if there's no family history. Building a life style of making good choices, addressing issues early and solving problems well is what you can model.
See below. Having two generations of major depressive disorder predisposes the child to depression. Symptoms tend to start off as an anxiety disorder. If one has a sibling with an eating disorder, this person is 10 fold more likely to develop the disorder than someone in the community.