"Fear them not therefore: for there is nothing covered, that shall not be revealed; and hid, that shall not be known."

Would appear to be a biblical justification for this view. However I suspect that other verses indicating that we won't be concerned about our lives on earth suggest to me that it won't work like this and that this verse relates to:

1) The process of judgement by God of all that the evil doers that Jesus is discussing in this passage

2) The reality that things DO come out in the end despite our best endeavours to cover them up.

On the other hand the challenge of testing everything we think about a person by what others, including the person, would think is one that can be helpful; if we couldn't justify it to them, what are we doing thinking like that...