First, look for any other solutions with 0.
If A=0 then the system becomes:
B*C = B^2
B = (B+C)^2
C = C^2
C is either 0 or 1. If C=0 then B=0, yielding the trivial solution (0,0,0). If C=1 then B = B^2 = (B+1)^2, which has no solutions.
Then the only solution (A,B,C) with at least one zero is (0,0,0).
If A=B then the original system reduces to:
A*(1+C) = (2A)^2
A*(1+C) = (A+C)^2
C+A^2 = (A+C)^2
The first two imply 2A = A+C, or A=C. Then A=B=C, which implies they all equal 1/3 or 0, but 1/3 is not integral.
To try to find remaining nonnegative integer solutions assume without loss of generality 1<=A<B<C.
The second equation states B + C*A = (B+C)^2 = B^2 + 2BC + C^2. But from the inequality B<B^2 and A*C<C^2. Therefore B + C*A < B^2 + 2BC + C^2 and the equality has no solutions.
The only nonnegative integer solution is (0,0,0).