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If this is what you mean:
[I made the parts I think you don't want on the left yellow, and wiped them out on the right, but they were part of the original Hatch pattern], then I don't know of a way to do that.
Maybe a way could be found to have a Polyline perimeter drawn that would step around the Circle following the "path" of the empty spaces between the squares, and to Hatch that, but it seems very complicated, since the nature of it would differ with every differences in Circle radius and/or Hatch scale. Also, assuming you want the squares to lie symmetrically in the Circles, the origin for the Hatch would presumably have to be calculated separately for every Circle, based on the Hatch scale, since the origin of that pattern is at the lower left corner of one of the squares, not [as I think would be more convenient] in the middle of one [or in the middle of the space between four of them].
EDIT: But here's a thought.... If you Explode such a Hatch pattern, then use PEDIT with the Multiple option and Join, and select all the results of Exploding, the complete squares will all turn into closed Polylines. The partial ones would also become Polylines, but open ones. A routine could quite easily be made that would delete all Polylines that are not closed from a selection.