Cub Scout nights of camping can (if they otherwise meet the requirements) be included as part of the BSA’s National Outdoor Awards for older Scouts.
“Under the auspices” is pretty broad:
Bryan on Scouting: Interpreting ‘under the auspices’ in National Outdoor Awards requirements
A “BSA unit activity” may include some individual efforts not part of the group: e.g., a pre-conditioning exercise where all unit members commit to and actually ride their bikes X miles a week prior to a big troop bike ride. It is not meant to include activities that the Scout undertakes that are unrelated to Scouting — e.g., family or church group camping, running as part of the school cross-country team, etc.
In some cases, an individual Scout could work toward a Scouting award and that could be counted at the discretion of the Scoutmaster. Such periods of time should be relatively short and focused on the award in question.
If the Cub Scout’s camping is related to earning a particular Cub Scout rank, adventure, or award, then I would include that in the camping log title and / or notes.
If the camping is just family camping and is unrelated to camping, then I would keep it in the logs, but add a note stating that it is personal / family camping and unrelated to Scouting.