I have eczema, but what happened to me after my second child, was not a flare up of eczema, it was something worse. And it sounds similar to what you have described.
I had thrush, on my face and arms. I was breast feeding, and my daughter developed thrush, I contacted a breast feeding consultant (instead of my doctor) because I could tell something was wrong with my breast, I had gotten the thrush on my nipples (Very Painful, like that first week of breastfeeding) She told me to continue to breastfeed, it wouldn't hurt. When my face wouldn't stop hurting, I went the doctor, the thrush had metastasized into my blood. The doctor said I should have stopped breastfeeding, we were only transferring back and forth. After five rounds of diflucan, my face cleared up but my breastfeeding days were over.
I have discovered that my daughter is VERY yeasty, for over a year, if she kissed me on my lips, my lips would break out again.