If you plan to "stealth" camp in the Whites, bring a hammock. The number of places you can set up a tent are extremely limited (legally or not) and the chances of finding one 10 miles from where you started in the morning is slim to none.
You are either climbing or descending steep trails with dense pine tree growth on either side mixed in with the boulders. Flat spots tend to be boggy. There are long stretches where camping of any kind is not permitted, like above tree line. Also keep in mind that there will be a fair number of thru hikers, section hikers and locals all looking to use those few stealth sites, which at most can only hold one or two tents. This is not something you can count on.
The good news is that most of the designated camping sites are about 10 miles apart.
If your starting at Beaver Brook shelter, you have to climb up and over Moosilauke first, which is one heck of a way to start a section hike. Then you need to climb out of Kinsman notch the next day, another dozy of a climb. That's a couple of rough first days. Hope it's not raining, or more likely, get caught in a thunderstorm.
The main problem will be between Eliza Brook and Liberty Springs campsites. It's only 11 miles, but will be 11 hours of hard hiking. You could stop short at Franconia Notch, but you can't camp next to the highway and the state campground is a few miles up the road. Unless you skip the AT and use a side trail which goes right to the campground, although you have to descend down a cliff to get there.
Lakes of the Clouds is pretty much a must stay. The dungeon only holds 10 people and is really rank. Maybe there would be a slot open when you get there, but probably not. Madison hut is also recommended. To go anywhere else means a big loss of elevation. Might as well go for Carter notch hut too to make that section a lot easier.