No. Identical twins are formed when one egg divides in two, leaving two eggs with identical DNA. Since an entire chromosome is different in girls and boys (XX vs. XY), they could never be considered identical. Twins of different genders occur when two different eggs are fertilized and implant themselves, which is why their genes are different.
Identical twins come from one sperm and one egg that splits in two after conception. It is impossible for them to be of opposite sex. Fraternal twins can be of opposite sex because it is two eggs and two sperm. I didn't see sillergoo's answer before I answered. It's basically the same.