Gordon,
There are tons of great places to camp in S. Utah that don't require a reservation....and where you won't have neighbors either. The trouble with Zion (and all the N. Parks) is you are competing with the masses. If camping shoulder to shoulder is your thing, then go for it. But if you want more room, then you have to get out and explore. Since you asked, at Zion there used to be a nice boondocking area just a few miles west of town, in the wash (watch the weather). At Brice, also go west (actually, east works too). Just before you drop into Red Canyon, there will be a little rest area on the south side of the road with a gravel road behind it that goes to a nice little forest campground. If that is full (there are only about four sites), cross the hwy and explore out the road that goes north.
You will find both of the above sites (and a lot more)
here.
Have a great trip,
Steve