What's that small red square in the upper left corner of my squad units in the Command Bar?

It means that you've issued HOLD FIRE from the Command Menu or that HOLD FIRE was initialized for those units in the mission you're playing.

When under orders to hold fire, your soldiers will not fire at will. They will still fire at their target if the FIRE, ENGAGE or ENGAGE AT WILL command are issued but will stop shooting as soon as the target is killed.

To revoke the HOLD FIRE command, issue the OPEN FIRE command.

To learn more about command a squad, look here.