I find myself quite often experiencing something similar, but not with broccoli. So far there are two (general) strategies that I find to work best when facing zombies.

If we take your zombie "If nothing really matters [ie, values aren’t objective, or however I put it back then], then it doesn't matter that nothing matters. If I choose to hold something as important, I can't be wrong."

1) Asking: What is it about "values are not objective" that I find so scary? If I could change that fact to something, what would that look like?

2) Questioning why the zombie is allowed make a statement with such emotional force, when it seems to imply that nothing deserves emotional force.

I'm not sure how much sens I make, but that's that.