Consider this system of equations:
A+B*C = (A+B)², and:
B+C*A = (B+C)², and:
C+A*B = (C+A)², and:

It is observed that A=B=C=0 trivially satisfies the above system of equations.

Does there exist any further nonnegative integer solutions to the above system of equations?

Provide adequate reasoning for your answer.