If I remember correctly that the U.S did switch sides between sides supporting both Iraq and Iran at some point due to some sort of Cold War Reason. So my question is did the U.S. back empowerment of both the Shah of Iran and Saddam Hussein?
The U.S. and Iran were originally allies when the shah they supported was in power. the Iran hostage crisis and subsequent revolution changed this by removing the shah from power and installing the Ayatollah as the new leader of Iran who was fiercely anti American.
Iraq was supported by the Soviets, however the Iran-Iraq War was taking place during the decline of the USSR and so Saddam was seeing less support from his Soviet allies and losing the war, so he was keen to switch sides and the U.S. was willing to help "punish" Iran for ousting the pro U.S. regime.
Yes the U.S. did support both, though they were at different times, the Shah of Iran was supported during the cold war pre-1980s when he lost power, and Saddam was supported by the U.S. in the mid 80s.