Let's break this down.
...registration Call to Action "component"...that encourages registration ... [and] could be repurposed in the header, footer, middle of a page - wherever you propose
Your design should be a piece of the user interface that can fit into many contexts and still do its job. Components/modules/controls/patterns are a central concept in design pattern libraries or design languages. Notice the section called "components" in Google's Material Design Guidelines. The design patterns they outline there are not simply cut-and-paste, but they show you (the designer) how to use a component that will be consistent with the design system they've set up.
This is the component's job. You could assume that it facilitates registration as well, perhaps, but it's important that it sells registration to users. You will probably want to do some homework on selling/converting vs. user experience, and be able to talk about the subject during your interview. Like DarrylGodden said in his answer, you probably don't want the component to interfere with users' other goals on the site.
"Full design mockup"
It sounds like they want you to deliver a high-fidelity comp (or maybe a prototype, for extra credit) of your design, and position it within a wireframe of the site for context. Ultimately, you could probably tell the story of your component however you'd choose, but know that they want something polished, not napkin sketches.
"Wherever you propose"
You need to propose where and how your component should be used by designers or developers. You might also propose how your component should not be used, if you think there's a chance that someone could misuse it.
TL;DR: You need to provide a high-fidelity mockup of a component that "encourages" registration, and show it in the context of the existing site (which doesn't need to be hi-fi). You need to show that you can think conceptually, and solve the problem in a way that can apply wherever that problem is encountered.