Camping Merit Badge Requirements - http://www.scouting.org/scoutsource/BoyScouts/AdvancementandAwards/MeritBadges/mb-CAMP.aspx
Camping - MeritBadgeDotOrg - http://www.meritbadge.org/wiki/index.php/Camping
9a. "Camp a total of at least 20 nights at designated Scouting activities or events. All campouts since becoming a Boy Scout or Varsity Scout may count toward this requirement. One long-term camping experience of up to six consecutive nights may be applied toward this requirement. Sleep each night under the sky or in a tent you have pitched. If the camp provides a tent that has already been pitched, you need not pitch your own tent." 
Ask the Expert: Interpreting Camping merit badge Requirement 9a - Bryan on Scouting - http://blog.scoutingmagazine.org/2012/06/07/ask-the-expert-interpreting-camping-merit-badge-requirement-9a/
Ask the Expert: What is (and what isn't) a camping night? - http://blog.scoutingmagazine.org/2015/06/24/ask-expert-isnt-camping-night-camping-mb/
MeritBadge.Net • View topic - Camping Merit Badge - Nights requirements - http://www.meritbadge.net/phpbb/viewtopic.php?f=10&t=3148
Camping merit badge- requirement 9a (2) - 137725 - http://www.utahscouts.org/document/camping-merit-badge-clarification/137725
Camping Merit Badge - Camping-Merit-Badge.pdf - http://rwstroop61.org/wp-content/uploads/2011/01/Camping-Merit-Badge.pdf
20 Nights Key Points
Here are some key points: 
- The requirement begins with “Camp a total of 20 nights.” That means 20 overnights, so a weekend trip from Friday through Sunday is two nights.
- Next it says, “at designated Scouting activities or events.” This means the experiences are held under the auspices of some level of the BSA, and that “Scouting” happens on them. For example, an individual family or a couple of Scouts and their parents heading off into the woods doesn’t count.
- A long-term camping experience is defined as at least five consecutive nights. One of these experiences is allowed, and up to six nights may count toward the requirement. For example, Sunday through Saturday. If a Scout goes on a 10-night trek, only six of those nights counts.
- If a Scout goes to summer camp twice for a total of 12 nights, only one of the summer camps will count — for up to six nights.
- The remainder of the camping nights must be accumulated through short-term camping — normally weekend troop campouts. The long-term camping experience must also be a “designated Scouting activity or event.” This could be at a council summer camp or on a troop’s own 50 miler, a Jamboree, high-adventure base, etc.
- All 20 nights must be spent under the sky or in a tent, so nights in cabins don’t count.
- If camping is done at a camp that provides tents that are already set up, then all is good. If tents are not already pitched, the Scout must pitch his tent. If he is sleeping in a two-man tent, then it would be reasonable the he and his buddy set the tent up together. Sleeping in a tent that Dad or the Scoutmaster, etc., pitched doesn’t count.
- I’ve heard feedback from parents with Scout in troops that don’t do very much camping. They can get in the long-term outing, but it takes a long time for their troop to get out on enough campouts to make up the other 14 nights. As a workaround they suggest they will send their son to summer camp, but then take him home after four nights so the experience will not count as a long-term camp. This doesn’t fulfill the requirement. The short-term campouts provide variety in both preparation and experience, and the Scouts are more likely to have to set up their own tent and take more responsibility for outdoor living skills. A long-term summer camp is still a long-term camp even if the Scout is there for only a portion of the time. It’s an entirely different adventure and usually doesn’t call for the same level of self-reliance required for a short term camp.