I estimate most people would lump "don't believe everything you think" into the space occupied by slogans like "think different" and "question authority"; i.e. at best a generalized endorsement of counterculture ideals, and at worst a cynical attempt to break down any and all ideals in hopes that the gap will be filled by something more congenial to the speaker. The general population is familiar with ideology and unfamiliar with abstract cognition, so unqualified ideas about ideas will usually be taken to refer to the former.

This misconception could be dissolved with half a minute of explanation, but that half minute wouldn't fit on a bumper sticker.