Look for a new doctor. If there are risks for getting pregnant when on Lexapro, she is probably worried about a lawsuit, but frankly she has no business making your boyfriend sign anything. Maybe she just doesn't want to have patients who are on Lexapro.
If you are angry about this, you could talk to a lawyer that specializes in malpractice. Your doctor really crossed the line, I think, and perhaps a letter from your lawyer would stop her from doing this to other people.
Besides the Big Brother aspect of this that is offensive, she has told you that your birth control must be the Pill when you could just as easily use an IUD, and is trying to bring your boyfriend into it even if you might change boyfriends sometime. So not only is the approach heavy-handed, it is absurd. What she DOES have the right to do, is ask you (not your boyfriend) to sign an informed-consent document about Lexapro. Any doctor would have the right to do that.
If you otherwise like this doctor OK, and don't want to go the confrontational route of contacting a lawyer, just tell her office that you and your boyfriend are not a unit in the eyes of the law and you'd be glad to sign an informed-consent document yourself.