We should pay closer attention to the narrative context as we read the text from which these two are found, before we make application: that is, hearing the significance, then making application.
Taking up one’s cross is often used devotionally to move Christians toward obedience; and, losing-finding one’s life is often used toward the nonChristian to draw them to Christ. Not bad things (or outcomes), but rather than a general (privatized) application of these–which, by the way, these are toward Christians for faithful perseverance and endurance in the faith, which is the context, e.g., “. . . the one who endures to the end will be saved,” 10:22b–and, thus, a more specific application is warranted.
First, the context is specific:
“For I have come to set a man against his father, and a daughter against her mother, and a daughter-in-law against her mother-in-law. And a person's enemies will be those of his own household. Whoever loves father or mother more than me is not worthy of me, and whoever loves son or daughter more than me is not worthy of me” (Matthew 10:35-37).
This makes better since of Jesus’ words: “Whoever finds his life will lose it, and whoever loses his life for my sake will find it” (Matthew 10:39). And, so, taking up one's cross (v. 38a) is related to what happens when following Jesus puts one in a place, “for Jesus’ sake” (v. 39b), where one can no longer count on the social construct available for life (literally for survival).
So, the significance of this text and these “clichés” is, following Jesus (which this text is about; v. 38b) means to die to (i.e., no longer count on or trust) the ways in which our social constructs and cultural norms provide identity, security, and worth. The Christian’s identity, security, and worth is located in Jesus and his cross. This is the gospel. This is truly Christian discipleship, that is following after Jesus. This is losing one’s life (i.e., giving up, abandoning, rejecting, not trusting in such social norms--again--for identity, security, and worth) to find real, true life through following Jesus.
The question would, then, be what social constructs and cultural norms give us identity, security, and worth? These we need to die to in order to have real, true life. This is what it means to actually follow Jesus.