Strictly based on number theory learnt in GMAT alone, I don't know a quick way to check perfect squares. However, based two special number properties, here's a fast way to do it -

1. Perfect squares always end in 1,4,5,6,9. Eliminate choices (C) and (E).
2. Digital roots of a perfect square are from {1,4,7,9}. Digital root is the sum of all digits of a number.
1+2+6+6 => 15 => 1+5 => 6(Not a perfect square)
1+4+4+4 => 13 => 1+3 => 4(Could be a perfect square)
4+0+3+4 => 11 => 1+1 => 2(Not a perfect square).

Option (B) 1,444 is a perfect square. Its square root is between 30 and 40. It could be either 32 or 38(last digit =4). Since, 1444 is closer to 1600 than it is to 900, 38^2 = 1444.