not sure if this is exactly what you want but usually least squares refers to fitting a line/function to a dataset, which doesn't really match your request.
I think what you're after is more like the center of mass (c.o.m.), which is given by the relationship (positions are vectors, so you'd need to do this for x and y seperately)
(total mass) * (position vector of c.o.m.) = mass1*position1+mass2*position2+....massN*positionN
you can solve for the center of mass (c.o.m.) position by dividing the right hand side by the total mass. masses act as your "weighting" and theres no reason you can't give them a negative value, although the effect may be different than what you're looking for, i'm not sure how else you'd define the effect of a negative weighting in something like this.