(apologies if this double-posted)
It’s my position that “four meters squared” is ambiguous, and can mean either an area of (4 m)[sup]2[/sup] or 4 m[sup]2[/sup].
I agree with zut’s point that “x meters square” refers definitely to a square that is x meters on a side. And square meters is similarly clear. But in my experience, it is quite often the case that “x meters squared” is used to refer to x m[sup]2[/sup].
Most of the time the meaning is clear from context. For example, if someone writes, “168 m[sup]2[/sup]” and then refers to it as “one hundred sixty-eight meters squared”, it’s not confusing. Written out in words, though, it isn’t clear. I suppose in terms of the bet this means that there is a difference, but one could mean the same thing, depending on context. Regardless, it’s better to avoid ambiguity and use some clear form (like “one hundred sixty-eight square meters”).
Here’s a link to someone saying basically what I just said : Ask Dr. Math : Meters Squared vs. Square Meters
The response quotes from the NIST stylistic conventions (which kind of skirt the issue a little bit), although his(?) quote leaves out a line that supports the use of squared to apply to units :