Put it differently, in an ungloved "street fight" would you dare to punch someone in the face with the fist? I wouldn't. All he has to do is to put his forehead into it and blam my fingers broke. The palm strike feels safer. Is it?
While punching someone in the face or forehead is not always the recommended way by a lot of people on here, there are some things you can do to mitigate the chance of your damaging your hands.
Spend sometime strengthening your wrists. Knuckle pushups are good exercise to help with this, with the added bonus of forcing you to align your wrists properly for when you punch.
Spend time punching a bag with no gloves on. At least get a feel for how to punch without gloves and to further toughen the hands up. It would be silly to completely replace your practice with no gloves practice, as that would detract from what you are trying to do kickboxing, but at least try to get in some time doing it. Get your knuckles and hands used to hitting things without that padding to protect them.
Here's the big one for saving your wrists: Never, ever, punch someone in the head with a horizontal punch when you aren't wearing gloves. Why do you ask? It is literally impossible to align your wrist properly for a punch above a certain height. Allow me to explain:
Walk close to a wall and extend your arm like you had just done a punch, wrist properly aligned, and keep the first horizontal. Keep you arm at a slight downward angle from parallel to the ground. and set your punching knuckles against the wall.
Slowly raise the angle of your arm. You'll notice as you reach near parallel to the ground, you can not physically keep your knuckles on the wall without bending your wrist out of the correct alignment.
Perform the same test, but with a vertical punch. You'll notice you can now keep your punching knuckles on the wall and have you arm well above parallel to the ground.
The take away from this? You risk hurting your wrist if you try to punch someone in the head with a horizontal punch. Gloves and taped up wrists help to mitigate some of this, but if you aren't wearing gloves you increase the risk of damage by a lot.