I always thought a point had a position but an infinitesimal size and a line segment had a way to specify the segment (start/end positions or other) but an infinitesimal width. Either way, that means there's an infinite number of points on a line segment and an infinite number of points (and line segments) in a square.
RE: Are there more points on a line or in a square? - Intresting bits of mathematics - week -2