The list is given in arithmetic progression format, not magic square format. If you want to convert formats, use this:

A B C        F A H
D E F  ----> G E C
G H I        B I D

Each AP is followed by its (m,n,t) key number generators.
That is, for the AP (A,B,C) = (a^2,b^2,c^2), where A < B < C
  a = |m^2 - n^2 - 2mn|t
  b = (m^2 + n^2)t
  c = (m^2 - n^2 + 2mn)t
with step value
  d = 4mn(m^2 - n^2)t^2

Note that t is the scaling value for the AP. If t = 1, then it is a primitive AP.

Some of the characteristics required to be compatible with a fully-magic square are:
(1) the t scaling of an AP cannot have a prime factor of 8k+3 (i.e., no entry can be a multiple of 3, 11, 19, etc);
(2) the t scaling of the AP that goes through the center cannot have a prime factor of 4k+3;
(3) the t scalings of the three APs must be pairwise coprime.

There are more characteristics, but this is enough to eliminate all but a few of the following list.

To be a fully-magic square of squares, the squares of the middle values of the three APs must form its own AP.

There are two groups of solutions below.
The first group is based on a search of all odd-entry APs primitive and scaled, for all d values up to d = 1.4 x 10^10.
The second group is based on an (m,n) search of primitive APs for all d values up to d = 10^19.

======================================
[d = 52492440]
 1367  7373 10337  (38, 77,1)
 5521  9109 11639  (55, 78,1)
17639 19069 20399  ( 5,138,1)
  Since all three APs are primitive, this is compatible with a fully-magic square.

[d = 71831760]
 2171  8749 12181  (12, 23,13)
 6227 10517 13507  ( 5, 28,13)
11633 14393 16703  (77, 92, 1)
25889 27241 28529  ( 4,165, 1)
  Any 3 of the above can make a semi-magic square, however, to be compatible, only one AP with t=13 can be used.  If both are used, then the three t values wouldn't be pairwise coprime.

[d = 322817880]
 11879  21539  28049  (31, 46,7)
 20489  27251  32641  ( 7, 62,7)
116273 117653 119017  ( 2,343,1)
  Not compatible. t values are not pairwise coprime.

[d = 1047566520]
 5739  32871  46131  (34, 99,3)
 8521  33469  46559  (85,162,1)
52467  61647  69627  (10,143,3)
  Not compatible. t values are not pairwise coprime and there is the 8k+3 factor, 3.

[d = 1627690680]
 27669  48921  63411  ( 15, 22,3x23)
 74589  84801  93909  (  2, 35,3x23)
134041 139981 145679  (259,270,1)
  Not compatible. t values are not pairwise coprime and there is the 8k+3 factor, 3.

[d = 2459457000]
43971 66279 82779  (22,147,3)
61295 78845 93145  (22, 27,5x13)
66885 83265 96915  ( 5,  6,3x5x7x13)
  Not compatible. t values are not pairwise coprime and there is the 8k+3 factor, 3.

[d = 3187938600]
 34705  66275  87065  ( 7, 34,5x11)
 74679  93621 109329  (34 ,41,3x11)
212295 219675 226815  ( 2,121,3x5)
  Not compatible. t values are not pairwise coprime and there are 8k+3 factors, 3 and 11.

[d = 6666332400]
 66615 105375 133305  (49, 68,3x5)
 83973 117123 142773  (25,196,3)
730289 734839 739361  ( 1,324,7)
  Not compatible. t values are not pairwise coprime and there is the 8k+3 factor, 3.

[d = 1.27 x 10^10]
122707 166673 201253  ( 70, 89,13)
159641 195469 225679  (285,338,1)
912899 919841 926731  (  1,266,13)
  Not compatible. t values are not pairwise coprime.

[d = 1.31 x 10^10]
196691 227579 254749  (95,108,11)
205343 235097 261487  (29,484,1)
248721 273801 296769  ( 4, 91,3x11)
  Not compatible. t values are not pairwise coprime and there are 8k+3 factors, 3 and 11.

===========================================
The following are all primitive APs and are therefore compatible.

[d = 8.81 x 10^12]
 2987849  4211981  5153089  (2035,266,1)
10522583 10933357 11329247  (3306, 61,1)
11330639 11713109 12083479  (3422, 55,1)

[d = 1.03 x 10^13]
 961241 3347261 4635119  (1610, 869,1)
6104543 6895333 7604327  (2622, 143,1)
6568753 7309493 7981783  (2002,1817,1)

[d = 3.58 x 10^13]
 1647887  6203957  8617577  (2201,1166,1)
 8120041 10085069 11725279  (2438,2035,1)
11258281 12748429 14081761  (3565, 198,1)

[d = 3.31 x 10^15]
10629079  58464869  81995759  ( 7238,2465,1)
60764257  83650813 101501833  ( 9077,1122,1)
93079487 109402717 123588503  (10434, 731,1)

No more found up to d = 10^19.