> Can I create header files with my own functions in java such as in C++?
Sure, but the java compiler will not do anything with them as it only compiles .java files, so technically you'd be wasting your time.
> If not, do I need always to make my own classes in order to call these functions?
Yes. A collection of utility methods is often provided in a class with a private constructor class with static methods. Then you can call them.