A simple explanation.
Squaring means multiplication, multiplication means repeatative additions.
Now if you add even no.s for odd no. of times or odd no.s for even no. of times you will always get an even no.
Hence, square of all the even no.s are even, means the last digit is always even.
- If you add odd no.s for odd no. of times you will always get an odd no.
Coming to the squares of odd no.s whose results are >= 2 digits. Starting from 5^2 = 25, break it as 5+5+5+5+5, we have a group with even no. of 5 and one extra 5. According to my point no. 2 the even group will always give you a even no. i.e. 20, means the last digit is always even. Addition of another 5 with 20 makes it 25, 2 is even.
Taking 7^2, 7+7+7+7+7+7+7, group of six 7's = 42 plus another 7 = 49.
Now consider 9^2, 9+9+9+9+9+9+9+9+9, group of eight 9's = 72 plus another 9 = 81, (72+9 gets a carry of 1 making the 2nd last digit even)
35^2 = group of twenty four 34's (1190) plus 35 = 1225, carry comes.
In short just check the last digit of no. that you can think of in the no. co-ordinate (Real and Imaginary) it will always be b/w 0-9 so the basic principle (point 2 and 3) will never change. Either the last digit will be an even or the 2nd last digit will become even with a carry. So the 1 digit sq can come odd, 1 and 9, as there is no carry. I have kept it as an exception in point 3.
BTW many, including the author may not like my lengthy explanation as mine is not a mathematical one, full of tough formulae. Sorry for that. I'm not from mathematical background and never like maths.