Unfortunately, this requires motion tracking and there isn't much out there that can handle it for free. (As Jason Conrad pointed out, DaVinci Resolve Lite can.) Motion tracking is pretty elaborate because it has to use complex computer vision systems to identify objects within the image and track where they move from frame to frame.
After Effects is probably the cheapest option on the list of tools that are often used for motion tracking, but it is still either several hundred dollars for a stand alone copy of the slightly older CS6 version or $20 a month to rent the newest version from Adobe.
With such a motion tracking capable tool, you can track the motion of the scene and then either stabilize it such that the eyes remain stationary or you can use the tracking data to have a rectangular mask follow the eyes and only reveal the contents of the mask. After Effects (or really any other program that supports this kind of tracking) is pretty advanced and gives lots of options for how to accomplish your goal, though you should also be aware that they are pretty advanced packages and there is a potentially significant learning curve.