I have come upon the equation $A^2+B^2+C^2-2AB-2AC-2BC$ and want to factor it. Because of the symmetry, I am wondering if it can be a perfect square or if there is some other nice factorization.
You can use Wolfram Alpha to get some alternative forms
The first two listed are $$(A-B-C)^2-4BC$$ $$A^2-2A(B+C)+(B-C)^2$$
As J.M. says, it will all depend on the value of $A,B$ and $C$
As is noted in other comments and answer, the symmetry is a little illusory because of the signs. This is probably less than you were hoping for, but or any integral choice of $B$ and $C$, you can find an integer $A$ which makes the expression a perfect square. Just take $A = 2(B+C)$, and the expression equates to $(B-C)^2$.