I agree with most comments, especially Toastmasters, but it sounds like you have another problem: You don't know your subject well enough.
You say you speak comfortably with your friends: compare the subjects you discuss with friends and your level of knowledge of them vs your what you know about your job. Do you know enough about the industry? Do you know what you're selling? What it's worth to the client? Strengths, weaknesses? Would you spend your own money on it? Why?
I've been in your situation before, where I barely could get a word in edge-wise even though I was expected to given my position. It was always my comfort level with the subject itself, not the setting. Imagine being in that same room and the client asks you a question about something you're intimately familiar with (your favorite topic, whatever that is). Could you answer confidently then? I bet you would.
If I had to guess I would say you need to do your homework more. You need to be able to anticipate questions and have good answers on hand. That will raise your confidence level. Internalize your company's industry and product as much as possible and then build out a plan for handling more responsibility in client meetings.
You can discuss your approach to meetings with your boss first. It would help to go over the subject matter in private (especially if it's new or constantly changing) and it would be helpful to decide which parts each of you will present.
You can also do this about Q&A: decide beforehand which areas you will cover and all questions on those topics will automatically go to you. Clearly defining your responsibilities will take a lot of the anxiety out of it.
I used to do this with my (quite senior) business partner frequently. We'd take the time to do trial runs in our own board room. It's not magic (meetings are still tough) but it helps.
If you are struggling, presenting a plan on how to handle meetings to your boss will also give you a lot of credibility. You're acknowledging that you need to contribute more and most importantly presenting a solution that will help you do that. If your boss is a good boss he/she will run with it and you'll both get what you want.
FWIW I've been through it before and came out on the other side. I'm a former marketing exec turned software engineer (the latter of which I'm infinitely better suited for) and probably share some of the personality quirks you've inherited. Before I got out of marketing completely I was considered a strong public speaker and could chair just about any kind of meeting successfully.
You're partway there by asking for help. Now you need to build out a plan and involve the people around you. It won't take long to improve the situation beyond that.