"I have a 3 or 4 year old dell monior not quite sure what model its one that is my girlfriends i never purchased the thing. It is hooked up to my late 2009 mac mini with a vga adaptor."
I'm wondering if the fact that the monitor is connected via VGA (rather than via DVI) could have something to do with this.
I'm not as well-versed in video connection schemes as I should be, but could it be possible that the particular VGA cable that you're using doesn't have the ability to transmit "sleep" commands or information from the computer to the monitor?
I'm guessing the "bouncing box" that you're seeing is what the monitor displays when it "loses" the input signals (which I believe is what happens when the computer sleeps, that is, it stops sending "color signals" to the monitor).
Can the monitor's menus be accessed to change this setting?
I'm using a Dell 1905FP display hooked to a PowerMac g4. Generally, I have my "sleep" settings set "way out there", so the computer doesn't attempt sleep for a long time (it won't go into deep sleep anyway, due to PCI cards installed). However, when it does drop off to sleep, this particular Dell monitor "goes black".
Perhaps the "bouncing box" is something more recent, purposely put there to show that the monitor is powered on, but has no "display input signal" from the computer.
The fact that the box "moves" will prevent any screen "burn in" or memory. If you can't stand the box being there, perhaps the best solution is to reach for the power button.
Another suggestion would be to try a DVI type connection.