How do I get past being disqualified from the military?

You have to understand they the military needs soldiers that aren't distracted during time or war. They are mostly concerned about how well you will hold up if you were in a near death situation would you be able to protect your men. Once the military doctor has denied you there is no other way around that. The fact that you will need to have prescribed medicine to take care of your eczema disqualifies you. I wish you the best and maybe you should try to serve our country in a different way.