Can two people use the results of their publication for their theses? I've been working on a case study paper with a colleague, who is the first name on the paper. He is writing his PhD thesis and I am writing my MSc. thesis under the supervision of the same advisor.

Is it okay both of us to use the same results/findings of this paper in our theses?

Advisor told me that this is not okay and added that even if the other guy would be okay with that, he didn't accept this.

At the time that I will be defending my thesis, the paper wouldn't be published yet. The PhD guy will be defending after me in the future.

What is wrong with that? People cite other people's work and use them in their theses. Why is it  not accepted for me to use the results of the paper that I am contributing?
This depends on your country and university regulations, etc. In general you can use the results of your papers in your thesis but e.g. in my case (Europe) if I would use a paper that I only co-authored I would have to indicate what I did myself and what was done by the other author(s).

On one hand it would not be understandable to me if you could not cite a paper in your thesis (and you should be able to also indicate that you are one of the authors and that you did this and that in the paper). Citing papers in other papers/books/thesis is something you should do to clearly state where the results you are using come from (unless it is a textbook stuff).

If you are about to copy-paste parts of the paper or the whole paper then yes, this may be different and could strongly depend on your university regulations and I expect you would be at least required to clearly state what you did yourself.

On the other hand it is probably never a good idea to go against your advisor directly. Ask him/her about this, say what would you like to do with the paper, hear them out in detail what is the problem with that and propose a solution. If this is a case of regulations then there is no going against it as you would only harm yourself. If this is the case of ethics then your advisor should and hopefully will explain the situation - that is also what advisors are for, or at least should be.

I would like to give you a better answer but at this point I really miss the background and big picture to do so.