Regret seems a very secular, and modern, emotion.
Regret is the desire to remake the present. To go back and take a different path.
But to repent is to acknowledge that we were wrong; a decision was wrong, a choice was wrong, a reaction was wrong.
Repentance does not, therefore, entail wanting to re-shape the present: It is possible, quite normal indeed, to repent past actions, yet be grateful for the present. Or vice versa.
What about the "Je ne regrette rien"/ "I did it my way" attitude which so typifies the modern Man?
If it was really about 'regret' then that would be fine - but the context tell us it is actually about repentance - the person is saying (in a Nietzschian spirit) I repent nothing.
They are saying: if I had my life over, I would do everything exactly the same all over again. Even knowing the full import and consequences - I would choose to re-live my life precisely, rather than any other possible life.
And that attitude is impossible to a Christian - is profoundly anti-Christian (which is of course why Nietzsche made having that belief the touchstone of existential sincerity).