"Birthday is just another day for me." expresses what you have in mind. In some contexts "another" can mean a new start (Tomorrow is another day), but "just another" never has that meaning. "Birthday is just any other day for me." doesn't sound very natural to me. You could say "Birthday is just like any other day for me.", to make it better.

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.

I would rather say 'Birthday is just a common day to me'. Another day implies a new start in English, it's not suitable assuming you want to convey the Chinese meaning of a simple day, no big deal.