My baby has eczema too. You don't have to give up breastfeeding. Eczema is actually very common among infants. Just ignore your 99 and forget about the breastfeeding idiot doctor. I am sure that breastfeeding is the best for your baby.
Don't give up breastfeeding!! It is the best thing in the world for an infant.
My baby has serious eczema on whole body even we provide breast milk. Then we start to think what kind of chemical around him induced the rash.
First we stopped using any shampoo and bath soap even we just used few drops a time. Then the eczema become a little bit better.
Second we stopped using wet wipes with chemicals. We only used "c c lok" and "DIY" to clean and say no to all other big brand. Then he become much better.
Last, we found that the only place on his body doesn't grow rash is his bum (pat pat). We asked the chinese doctor about this and he said urine is sometime good for skin. In chinese tradition people used a baby urine to cure his own skin problem (but don't use other's). Then we started to used his own urine to test on the skin from he leg first. It works! And then we trust this method and use the same way in other area on his body.
But remember, if there is wound or the skin is "broken (blood or any body fluid come out) you should NEVER use this method. Otherwise bacteria infection will occur immediately.
So you can try on the baby's leg or back (usually there are less wounds because the baby can't scratch it's back) first. If it doesn't work, stop it. If it works try on the other place and keep going. I just used the wet diaper massage on his body 1 to 2 time a day.
Now my baby still has little rash once a week or a month but it's much much better. We keep not using shampoo and wet wipe with chemicals.