No, I would not add up the Type III Sums of Squares for the denominator. That denominator has no real meaning.
How you calculate your "% Variance Explained" will depend on the information that you are trying to convey. Typically, people will use the Type I SS divided by the model SS since the Type I SS will add up to the Model SS. Your percentages will then add to 100%. This is a "% Variance Explained", but these results will depend on the order in which you put your model terms.
You COULD use the Type III SS divided by Model SS. That will result in a total percentage different from 100% (it should be less than 100%). In that case your result is really the "Additional % Variance Explained after accounting for all other model terms".
Because of these issues, I am not a big fan of determining the percent variance explained for each term. You can rank-order the importance of the terms in other ways, which JMP provides, such as the Effect Summary table which also shows the magnitude of the term's impact.
Dan Obermiller