If it was non-public activity, I could imagine a court finding that it is. I could also imagine courts finding using social media activity that doesn't unambiguously indicate intent to commit a crime as part of a firearms background check to be a violation of the 1st and 2nd amendments.
The government looking through your personal information requires due process, otherwise its a violation of the 4th amendment.
Likewise, if the first amendment said something along the lines of "A personal connection with his creator being necessary for a upstanding and moral citizen, Congress shall make no law respecting the establishment of a religion or prohibiting the free exercise thereof..." I think you'd also have a hard time arguing that athiests and agnostics don't have a right to free expression or to not have a religion forced upon them by the government.
Who knew the solution was so easy.
Checking your activity on social media during a firearms background check is not a violation of the 4th amendment.
You can certainly enforce the second amendment militia requirement, although it will require a vast different SCOTUS than the one we have now.