Yes use Conditional Formatting:
I assume that you want A1 background to start white.
Therefore set up two conditions in A1:
=B1="x" and choose fill red
second condition:
=C1="x" and choose fill green
You haven't however stated what you want to occur if B1 & C1 both contain x.
With the above formula conditions if both contain x, then A1 will turn red
Hope this helps