I think you're reading too far into the birthday message. I send messages like that to my guy friends, that I wish I could see them or that I miss them, and I have no intention of ever hooking up with them and it's well known between both of us. I generally tell people to trust their instincts and that if you are snooping through your SO's social media or emails or whatever, that you simply don't trust them and you will find something you don't want to see, because you will place meaning behind messages/interactions when there is no ill intent behind them. In this case, I think you should start doing some digging in your own brain and thought patterns and find out what the root of this problem is. Why don't you trust your boyfriend? Do you react like this about all of his female friends, or is this his only female friend, or is this the only female you feel threatened by? Has your boyfriend ever acted in a way that has broken your trust?
I think you need to start communicating more with your boyfriend about this, but try not to be so emotional about it. I don't mean hiding your feelings, but I mean that you should bring it up at a neutral time, and use non-accusatory, non-judgmental language when you bring it up to him. Talk to him about how you feel and why you feel that way, without freaking out about it. It may seem hard to do, but if you ever want to get past this, you're going to have to. Chances are he won't want to end the friendship with her just because you asked, nor should he have to. It's not up to you to control who his friends are or who he spends his time with. So you have 2 options: 1) Get over it and accept it for what it is and decide right now that you're going to trust your boyfriend; or 2) Break up with him. It doesn't sound like you want to choose option 2, so I suggest you try to get to know the friend. Set up a lunch or coffee date with both of them so you can see them interact with each other, or if you are comfortable, try setting up a coffee/lunch date with just her. To clear the air and to get to know your BF's friend.
I've been dating my boyfriend for 7 months and have gotten to know most of his friends, some of them are females. Knowing them makes it so much easier to qualm those fears that he might be into those girls; but honestly, unless they dated before you two got together, I doubt you have anything to worry about (and even if they did date prior, it doesn't necessarily mean anything will happen between them). What will destroy your relationship is your unnecessary behavior and attempts to control who he is friends with. That will ruin your connection and he won't want to hear you out when you try to talk to him about how you feel. I think you owe it to yourself and your BF to make more of an effort to understand where he is coming from, and to let go of some of the insecurities you are feeling. Trust your boyfriend until he gives you a reason not to trust him, and learn to let go of some of the fears you have because it will not do you any favors.
"Caring is not an advantage."