>Is every
rational point on the plane equidistant from two integer
points?
Yes. Let the rational point be P = (p/q, r/s).
Arbitrarily suppose that the midpoint of the integer points is the
origin. The line from the origin to P is the perpendicular bisector of
the line segment between the integer points, so the slope of the latter must
be -ps/qr. Thus the integer points can be (qr,-ps) and (-qr,ps).