Vertically centre page content

Site URL: http://johnlauriephoto.com/story/n-a-b/

I recently saw this website and I love how this slideshow works: Click onto a project, then an image slideshow comes up that is aligned centrally within a page, when the window is resized (best seen in desktop). 

Is there a way to do this or something similar in Squarespace?

I think your first stop should be a Portfolio page. Then for each Project within use a Gallery section with Slideshow Simple.

Take is slow to see if you get the effect you are looking for before uploading massive amounts of images.

Let us know how it goes.