The reason that it's not recommended (aside from the stretch factor) is that you want something in the 10x your weight break strength range. Most folks average 150 to 200 lbs, and Amsteel Blue 7/16" is rated to 1,600 lbs--1/8" to 2,500 lbs. That gives a comfortable safety margin.
The reason you want that safety margin is because it isn't just your weight acting on the suspension; any movement in the hammock increases the dynamic forces on the suspension. I'm not a physics or math guy, so I can't give you the exact equations, but it makes sense. Figure you have a beer can. If you just put thirty pounds (number drawn out of certain orifices; don't quote me on this) on top of the can, it's likely to take the weight. Now, if you drop that weight down on top of the can from a couple inches above it, it's likely to crush the can.
The other reason is stretch. You don't want to find out just how much nylon stretches--especially when wet--by winding up over some sharp object on the ground and ripping your hammock fabric. Not fun at 3 AM when you're miles from the trail head.
Anyway, those are the reasons. Theoretically, you could use any line rated to about 10x your weight. Amsteel just happens to be the most commonly available and lightest weight.
Edit: What pgibson said, too.