In this context, you would say "My birthday is just another day for me." This means that it is an ordinary day and not a special day.
"Any day" would be used differently in a number of different ways. If you said, for example, "Today is not just any day, it is my birthday," that would mean that your birthday was special. However, you could say also, "Today is not just another day . . . ." But you would not generally say "it is just any day." So, no, they are not interchangeable.