Question 1019875
The statement is not always true.  Take for example k = 5.  Then n = 4*5 + 1 = 21, but 21 is not the sum of squares of odd and even natural numbers.  (Another is when k = 8.)

The converse goes this way:
For all even natural number a and odd natural number b such that n = a^2+b^2, then there is a natural number k such that n = 4k + 1.  (And because a statement and its converse are equivalent, the converse is also not true.)