I can see how she could misunderstand what you said, but I mean, hey. It's her birthday and you're her friend, and I would want to hang out with my friend on my birthday too. So I mean, she' probably irked, but try to make it up to her by hanging out with her just the two of you another day.
If you told her you could hang out on her birthday, then I could see why she'd be upset. But thats what happens when you start working. Sometimes plans have to change. It doesn't look good to just skip out on a job on short notice.