He should be a fully fleshed, three dimensional character with more going on in his mind than daddy told him mutants are bad. Your mutants taking him hostage will initially reinforce such thoughts. The ‘good guys’ - if they exist - don’t usually kidnap people.
As Monica said, it is a gradual process and there will be setbacks.
One man’s freedom fighter is another man’s terrorist. It is a matter of perspective and degree.
I am playing with a scenario where an assassin is being persuaded to assist in at least investigating who might be ultimately responsible for the assassination he just committed. It is a very gradual process, one character refers to it as seduction and much like training a colt. Gentleness, consistency and logic slowly allow him to entertain the possibility of helping these people - in as much as honour and duty allow.
You want your character to suddenly realize he was wrong after more evidence has been presented that Dad was right. You need someone with outstanding people skills and a ton of patience.
Force will never work. One thing that you will probably need to show is that there is evil in everyone. Your most virtuous character has done something worth regretting. Shades of grey, not black and white. Black and white - aside from being a little naive - tends to be dull.
Day must make a choice for reasons of his own that are both unique to him and firmly rooted in his soul.
Consider Javert - his life spent chasing criminals and certain of the immutability of human nature. He encounters Jean Valjean several times, but seeing Valjean risk his own life to save another’s drove home how wrong he’d been, the damage he had wrought and that man can be redeemed. Valjean was redeemed when a saintly bishop bought his soul for god. The generosity of the act sent him in a tailspin, from which he emerged an altered man.
Valjean dedicates his life to helping others and raising the daughter of a woman his company wronged. Javert writes a damning letter to officials and leaps into the Seine.
How will your character cope with an awareness of all that he has done having been wrong? He probably uses the rationale that such deeds were necessary for the safety of the State.
Terrible deeds are done in the name of national security. My MC kills many people, believing it to be right and justified - until he receives orders he cannot reconcile with his moral code.
Have Day realize on his own that he had been duped. He might start by thinking ‘Not all mutants are evil. Maybe I should listen to these people - if they ever take the gag out - might ask a question too’.
Much of this will be determined by his age and level of maturity. If he is some callow kid following daddy’s orders because daddy said so, it can be sudden, but I probably would have put the book down not really caring if villain jr learns something.
Good villains are not easy to write. Even the darkest does not consider himself a villain - perhaps a tool of destiny - but never a villain. You have to learn everything there is to know about both Day and his father - not just the labels you assigned them. When writing them, be on their side. Love them and know the better your villain, the better your story. Do not make them cardboard cutouts.
Make him real, like him and get inside his head. Once there, ask him what it would take to turn him.