Let x^2 be less than some integer N. Has anyone proved whether or not there are solutions to y^2 = x^2 mod N for all x^2 <= N?

For example, if N = 27 are there solutions to all of these congruences:

y^2 = 4 mod 27

y^2 = 9 mod 27

y^2 = 16 mod 27

y^2 = 25 mod 27