Here is the answer that fulfils the conditions:--
A = 4 B = 3,364 C = 6,724 D = 2,116 E = 5,476 F = 8,836 G = 9,409 H = 12,769 I = 16,129
Each of these is a square number, the roots, taken in alphabetical order, being 2, 58, 82, 46, 74, 94, 97, 113, and 127, while the required difference between A and B, B and C, D and E. etc., is in every case 3,360.