A theorem exists in a book but not in a journal - is this considered as a publication?
First, you should take pride in proving a theorem that was significant enough for someone to attach your name to it. Even if it was "not very important", as you say, it was still enough to be worth mentioning. Not many undergraduates achieve that. 

When you submit to a journal, you don't submit a theorem, of course -- you submit a complete paper. A typical paper has more than one theorem, unless the one theorem is particularly good. In this case, if you describe your theorem as nontrivial but "not very important", it seems unlikely to make a paper on its own. 

If you write a paper, you could certainly include the theorem in question. Because the original book attributes the theorem to you, everyone knows it's your theorem, so there is no issue with 'stealing credit'. In the paper, you could say something like


  The following theorem has appeared in Smith [1, Theorem 4.5].


just before you state and prove the theorem.  This sort of thing is routine, and as long as the theorem fits nicely into the paper it is unlikely that the referee will complain.  To answer part of the question, you should mention that the theorem has appeared in the book already. 

At the same time, you will need to make sure that the new content of the paper is enough to merit publication (that is, the previously unpublished results that you put in your paper).   As you read more papers in your area, you will get a better sense of how this works in practice. 

I think you have also asked whether you can include the book in your vita. The answer is no. But at various times you are asked to write a research statement or research narrative. Because the book attributes the theorem to you, you can state and take credit for the theorem in your research statement. This is particularly relevant if you are applying to graduate schools.