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equidigital n.  and  square n.

First numbers that belong to both sets:

1, 16, 25, 49, 64, 81, 121, 169, 289, 361, 529, 841, 961, 1225, 1296, 1600, 2025, 2304, 2500, 2916, 3136, 3969, 5184, 5625, 6400, 9604, 10816, 11236, 12321, 13225, 13456, 13689, 13924, 14161, 14884, 15129, 15376, 16641, 17689, 17956, 18496, 19881, 20164, 21025, 21316, 21904, 23104, 23409, 24025, 24964, 25281, 25600, 25921, 26896, 27556, 29241, 29584, 30976, 31329, 31684, 33489, 33856, 34225, 35344, 36864, 37636, 40401, 41209, 42025, 42849, 43264, 44944, 45369, 46225, 47089, 47961, 50176, 53824, 55225, 55696, 56169, 58564, 59536, 61504, 62001, 67081, 68121, 70225, 71289, 71824, 73984, 75625, 77841, 80656, 82369, 82944, 84681, 85264, 87025, 87616, 88209, 90601, 92416, 93025, 99856, 101761, 103041, 104329, 106276, 106929, 111556, 112896, 114921, 116281, 119716, 122500, 123904, 128164, 129600, 131044, 142129, 142884, 145161, 145924, 148996, 152881, 154449, 155236, 158404, 162409, 163216, 165649, 168921, 169744, 173056, 173889, 178084, 183184, 190096, 190969, 198916, 199809, 202500, 203401, 204304, 205209, 206116, 209764, 217156, 221841, 223729, 228484, 231361, 232324, 239121, 240100, 243049, 251001, 252004, 254016, 255025, 258064, 264196, 265225, 267289, 269361, 274576, 275625, 276676, 277729, 284089, 286225, 288369, 289444, 291600, 293764, 294849, 295936, 297025, 300304, 303601, 306916, 309136, 312481, 313600, 315844, 319225, 320356, 328329, 335241, 339889, 343396, 345744, 346921, 349281, 355216, 356409, 360000, 364816, 369664, 373321, 376996, 386884, 391876, 394384, 395641, 400689, 401956, 403225, 421201, 425104, 429025, 438244, 444889, 446224, 447561, 450241, 454276, 463761, 469225, 471969, 474721, 478864, 481636, 483025, 485809, 487204, 488601, 490000, 494209, 495616, 498436, 499849, 508369, 512656, 514089, 515524, 518400, 519841, 522729, 524176, 534361, 538756, 540225, 541696, 543169, 555025, 556516, 561001, 562500, 567009, 570025, 571536, 574564, 582169, 583696, 586756, 588289, 594441, 595984, 605284, 606841, 609961, 616225, 620944, 622521, 625681, 628849, 630436, 633616, 638401, 643204, 644809, 651249, 652864, 656100, 660969, 664225, 667489, 669124.

Some larger examples :