The eczema isn't transmitted sexually, but your cold sore could be. That's herpes simplex type 1, and you can transmit that to your boyfriend's penis, giving him genital herpes type 1. If he already has hsv1, then you don't need to worry about it.
Lots of people have it and don't know they do, so he can ask his doctor for a type specific IgG blood test to find out.
If you aren't getting treatment for your eczema, you should talk to your doctor, since it sounds like it's causing problems and triggering cold sores for you.