It's all already been said, and quite well I might add. I agree with those here who said something along the lines of being able to train your mind to see things differently, in more positive ways, which ultimately brings happiness. But if you're asking if you think someone can just say to themselves, "I'm going to be happy," and voila they are happy? Well, yes, if they have the right mindset and personality I think.
For those who have struggled with very negative thoughts for a long time, and feel bad about themselves or their lives in general, I don't think you can just choose to be happy suddenly if you don't feel it inside. Then you are just lying to yourself. If you are happy, you are happy. If you're not, you're not. The question is: WHY? What can you do to make positive changes in areas that will contribute to your overall happiness?
I will say though that I don't think this means that everyone can be happy if they just set their mind to it. There are people in this world who deal with some horrendous things, and while it certainly isn't IMPOSSIBLE for them to be happy, how can you blame them if they don't feel content? Some people can only take so much and their positivity runs out.
I myself deal with constant anxiety daily. Just about every social interaction I have involves my muscles becoming tense, my breathing shallow, my nerves feel like they are stretched tight. This has always been the case. I always feel awkward, even in the most friendly, laid-back exchange. My mind seems to perceive every little thing as a threat. Anxiety in and of itself is a negative feeling - it doesn't feel pleasant. If someone is experiencing this constantly, and then subsequently feels unhappy quite a lot, I don't understand how you can blame them...? Those reactions can be changed and I work on them daily, but it takes a hell of a long time, and the fact is it may never go away completely - it is documented that those with social anxiety in adulthood almost never completely get rid of theirs (I don't have a source right now, sorry, but if I find one I will post it), but they can minimize it so it is manageable. I think mine is manageable at this point, but it is still depressing.
I guess one way you could be happy while having SA is just accepting that is the way you are and be more understanding with yourself, but it's understandably hard not to long to be at ease in a conversation. No one likes the fight-or-flight feeling, that is not a very positive way to live your life, constantly on edge.
Obviously the root of this anxiety are beliefs we hold about ourselves and others, but how to change them? Well, if I had the answer I wouldn't be on here now would I?

But it may not even be just beliefs. Some scientists believe that people with SA have something off in their brain that perceives certain things as threats that socially "normal" people do not; they think ours are tuned to pick up on little details that others do not, such as a subtle tone (OH NO! SHE DOESN'T LIKE ME!!! I CAN TELL FROM HER TONE!). I don't have a source but if I find one I will post it. I distinctly remember reading that piece of information, and it made perfect sense to me.
Sorry for the long post.