Each of these two estimates (DZ correlation and MZ correlation) express a numerical relationship (Pearson correlation) between individuals. Each of these could independently be squared to provide r^2 values, if that were one's goal. And the final quantity (heritability, h^2) that Falconer's formula provides is analogous to a r^2. But that does not mean that all of the intermediate steps can be easily treated in terms of r^2.
I'll defer to wikipedia and other resources for more details, but in simple terms, variance (in r space) is easy to decompose into contributing factors (e.g. MZ correlation vs. DZ correlation), but r^2 is hard to decompose into contributing factors. See for example the general definition of r^2 given by wikipedia:
When you are computing R^2, you don't sum (or subtract) variances explained by different factors. You compute residuals (differences like MZ vs DZ), square them, and sum them, and then say all the variance that isn't in that residual quantity must come from the model (e.g. is variance explained by the model).
I think one issue here is that people use terms like "variance explained" in slightly informal ways. It applies to both r^2 and the final heritability estimate, but not every intermediate step of the computation of those estimates can be neatly phrased in those terms.
For more background here, I can recommend the wiki page for heritability.