Thanks for the info, Peter. That would have been a real game-changer for local history–to have that house as KHS headquarters. My mother was born in The Bronx in the 40’s and she has VERY strong feelings about Robert Moses (mostly about the Cross Bronx Expressway). Yet another reason to be angry at Moses.
Yes, my understanding is that the house stood where there is now a utility shed/bathroom building by the baseball field. The below animation shows where it was. There is a contemporary Google map there showing the park, an 1885 map with the Johnson house labeled, and a map that I made depicting the area during the Revolution (The red square on the map represents the Berrian House).
If you want to fool around with my map program the link is here (although it is still a work in progress).
Here you can see the house on a map from the Revolution that is in the collection of the Library of Congress: