Well, I'll advice you to memorize squares upto 30. Then apply these techniques to find squares quickly . The process is best explained through an example.
1) Squares for Numbers from 31 to 50 : (Trick : Treat as 50 -x )
Suppose you need to find the square of 37.
Process A : Then 37 = 50 -13
now (-13)^2 ie square of 13 = 169 [ Here take 69 as last two digits of square of 37 & carry over 1 to process B.
Process B: First Two digits are derived by
25-13=12 + Carry over from process A
ie First Two Digits = 12+1 =13
Hence (37)^2 = 1369
2) Squares for Numbers from 51 to 80 : (Trick : Treat 50+x)
Suppose you have to find square of 67. Then 67 = 50 + 17
Process A : (Last Two Digits Finding)
17^2 = 289
Hence Last Two Digits = 89 [ 2 will be carried over to First Two Digits ]
Process B: (First Two Digits Finding)
First Two Digits = 25 + 17 + 2 =44
Hence square of 67 =4489
3) Squares for between 81 to 100 : (Trick: Treat 100-x)
Suppose we have to find the square of 82. Then 82 = 100-18
Process A: (Finding Last Two Digits)
(-18)^2 = 324
Last Two Digits = 24 [3 will be passed to First two digits to add]
Process B: (Finding First Two Digits)
First Two Digits = 82 (whose square we have to find) + (-18) + 3 = 67
Hence square of 82 = 6724