WASHINGTON—Supreme Court justices pushed back Wednesday against government claims that police should have broad access to data collected by telecom and internet companies about customers, with justices across the ideological spectrum suggesting constitutional safeguards must grow to meet threats to privacy and property rights posed by the digital age.

“Americans, I still think, want to avoid Big Brother. They want to avoid the concept that the government will be able to see and locate you anywhere you are at any point in time,”...