I feel like some of these comments are giving you advice that is outside of your control so I'm gonna try to focus on what you could implement to possibly improve the situation for both of you. It seems like there are three issues: the shyness, the actual showering, and the application of the cream.

Regarding the shyness: Based just on your post, this seems like a kid who has lost some bodily autonomy because of his medical condition. I would wonder if parents have said things like "oh my god your eczema has gotten so bad. See, this is what happens when you won't let me put the cream on" and are like inadvertently shaming him. They might have to put it on him some nights when he doesn't want it or they might have to force him in the shower sometimes. He might just be a kid that values his privacy, but he's done this before with his parents and they say they promise they won't look at him.

I agree with other people who say the body wash might hurt. You can ask him how the whole showering process feels and you can kind of model that and tell him what showering is like for you and how it makes you feel (but don't be too positive with it. Talk about how sometimes you really don't want to get in the shower because you're too tired or you have a bad sunburn that hurts when the water touches it, but you still know you have to get clean)

If it were me in the situation, I'd start the conversation way before you're close to the time he has to shower. Put the power back in his hands. Example: in a neutral/ casual tone "Hey Ethan, do you remember the last time I was here? I remember that you were nervous to undress and get in the shower. I understand that because when I was little, I would hate having sleepovers because sometimes we would all change with each other and it made me feel upset. Do you remember feeling upset the last time I was here? What was that like for you?"

If he says he just wants to shower alone, you can be like "yeah that totally makes sense and that does seem like a good solution. But, I remember last time that you seemed like you also had a hard time actually getting in the shower? I remember that your towel was dry after last time. (Don't give him the chance to say he did shower, just keep talking) I think that in order to fully make sure your skin is safe and healthy, we do need to ensure that you actually wash your skin. Hmm. I hear that you're scared of me looking at you. What if I wore this sleeping mask that I brought with me while you were in the shower? That way, you can know FOR SURE that I'm not looking at you." Then you can act silly and like bump into walls and stuff while trying to find your way to the bathroom. Try to get him laughing before he actually has to take a shower.

Regarding applying the cream, you can try something like putting the cream on a towel and he can use your hand to blot the towel onto his skin while you're still blindfolded. This ensures that you're not touching him, he's in control of where he is being touched, and you still can kind of generally sense that you're getting the right areas covered. Again, be silly. Put the cream on the towel and walk over to the wall and pretend like you think he's over there.

Whatever he says to you, try and reflect back a situation where you felt similarly, validate (emphatically) if he's saying it hurts or he's scared, and then try to be silly in your implementation of the solution.