When you create an MVP, do you code the entire back end?

I don't think there is any "one size fits all" approach to MVP's. Keeping in mind that the exact boundaries that define both "minimum" and "viable" can vary based on the context, there's a pretty wide range of outputs that can be an MVP. I'm pretty sure that somewhere, out there, somehow has used an MVP that used no code whatsoever, but rather just UI mockups, a powerpoint deck, etc.

Keep in mind what the goal of an MVP is: it's a tool to help you validate (or invalidate) a hypothesis. What hypothesis? That depends. On you, your idea, what stage you're at with the idea, yadda yadda. If you're trying to validate a hypothesis like "There is at least some interest in a product like X in the general sense" (for whatever your X is) then maybe UI mockups and/or a pitch deck are all you need. If the hypothesis is "if we build Z, then people will pay Q dollars a month to use such a site" then maybe you need a working product, which just doesn't have every bell and whistle you might eventually add, but which fundamentally captures the core of the idea.

And so on.

That said, for the products I work on, if there's a backend element and I need to code it, I am almost certainly going to be using either Groovy or Java for most things. If I'm using Groovy that generally implies that I'm using Grails for a web app and if I'm using straight Java then that probably implies using Spring Boot for a standalone service of some sort.

One of my friends is using a no-code backend for an MVP. They want to rewrite everything after the MVP stage and find a few clients.

Why wouldn't "you use some tools that speed things up?"

Good luck.