The answer is yes, at least for $2$-fiber products. And fortunately there is an excellent reference online : this is Lemma 52.8.4 in De Jong's stack project. I quote :
"Lemma 8.4. Let C be a site. Let f : X → Y and g : Z → Y be morphisms of fibred categories over C. In this case the stackification of the 2-fibre product is the 2-fibre product of the stackifications."
To get there, and get the proof :
http://stacks.math.columbia.edu/search?keywords=stackification
then select the first answer Lemma 52.8.4 and in the .dvi look for Lemma 8.4. Enjoy !
ps : the permanent tag is Tag 04Y1
