Don't waste your time with unpublishing plugin or module or any component. Because this is 100% related to Joomla Core. Looks like you are newbie to Joomla.
Your 2 screenshots use different "Menu Type"(s). As you can see, one of them is: "Single Article" while the other is "Featured Article"... They are different types, different concept.
I don't actually understand what you are trying to say though. I guess you are using wrong terminology. You are mentioning "module" but i don't see a module in the screenshots you have provided. You can assign Modules to MenuItems from the "Module Configuration Page",
not from the "MenuItem Configuration Page".
One way or another, this problem you are having problem, has ABSOLUTELY nothing to do with our template and unfortunately we don't provide free support or step-by-step instructions if you are asking help for non-related to our template inquiries.
You may purchase premium support:
www.flexiblewebdesign.com/premium-support
and let us take care any kind of modification or implementation for you directly on your website though.
Regards.
Support