What you need to do, is check your monitor settings for how it deals with different aspect ratios. You're looking for a 1:1 setting.
Keep in mind that this may affect any other inputs used by that monitor. Sometimes monitors are capable of having per input settings, other times it's a one size fits all. So you may end up having to change it back at some point.
Of course the 360 should be capable of at least 1080p. The only reason you'd get less than that is if your monitor weren't HDCP compliant. It's also possible you may simply have to use the VGA adapter for that. In which case, I'd point you to the Monoprice website where they sell a VGA adapter for the 360 for like $20 I think. I have it, used it for a while before I got a TV with HDMI, and it works great.
I have a Gateway 24 inch monitor which i use for my computer, i have my 360 hooked up via hdmi, the monitor native resolution is 1920x1200, the 360's max resolution is 1360x768 i think, and thus when i play the image is blurry cause of the small resolution on a big screen(duh) there a way i can just make my monitor/screen ONLY show the max resolution instead of using the 24 inches and making the quality crap,i dont care if their is a huge black box around it i just cant stand the current quality. basically i want to have the 360's max resolution WITHOUT stretching it to fit the screen.
http://www.gateway.com.au/systems/product/529668253.php?recid=529668253&source=/systems/product/529668253.php&series=529598106&industry=&related=& consumer=1.10.new&tab=specs#prodBar
my monitor

Chowhound
Comic Vine
GameFAQs
GameSpot
Giant Bomb
TechRepublic