Everyone proposing clever technical solutions. Burner phones and encrypted cloud data etc... you are solving the wrong problem. Instead of engineering a way for the few people who care deeply to avoid this at great effort, engineer a way for this to not be legal. Write your congressperson, donate time and money to political parties, make privacy at the border for ALL (not just citizens) a priority in every election.
the social and political aspects of it are not. there's no political traction against this stuff. the border patrol types are too stupid to understand simple and innocent explanations, and treating them as equals is a surefire way to incur their wrath.
the way to make privacy at the border work for everyone is for everyone to conspire to keep the border patrol in the dark. throw them a (fake) bone or a (true at first evidence check) lie, and keep moving on.