1. Let the number you're after be p. There are only 4 possible senarios in modulo 4.

p = 0 (mod 4) => p^2 = 0 (mod 4)

p = 1 (mod 4) => p^2 = 1 (mod 4)

p = 2 (mod 4) => p^2 = 4 = 0 (mod 4)

p = 3 (mod 4) => p^2 = 9 = 1 (mod 4)

Therefore any square number would have the property that p = 0 or 1 (mod 4)

2. As you can see, you can write each of the 1111.....11 (n digits) numbers as:

(n-2) digits ---> 111...111 x 100 + 11

In mod 4, this is the same as 111...111 x (4 x 25) + 11 = 11 = 3 (mod 4)

Since 111...111 is not equal to 0 or 1 mod 4, it cannot be a perfect square (from 1)