You can choose any square numbers to start with but you'll find that only certain combinations of odd and even numbers will work. You'll need to solve three simultaneous linear equations in three unknowns to find formulae for the three integers in terms of the three square numbers that you started with.
For the second part you'll need a number which can be written as the sum of two squares in at least three different ways. You could use this example:
$1^2 + 132^2 = 28^2 + 129^2 = 36^2 + 127^2 = 55^2 + 120^2 = 63^2 + 116^2 = 80^2 + 105^2 = 17425$