It is.If it's not “per meter squared”

I think you might be over-thinking this a bit. Units are a lot like variables, in terms of their algebra. I can take meters plus meters and get meters. I can divide meters squared by meters and get meters. I can only add two quantities-that-have-units together if they have the same units (just as in adding like terms with powers of x or y). Finally, there are certain functions for which the argument must be unit-less: the exponential function, the trig functions, the logarithm function, and others.