A couple of things.
It helps a *lot* to back the fabric with freezer paper. Ideally you want to iron freezer paper to the back of the fabric before cutting the squares. To ensure that guests do not sign into the seam allowance, you then want to add blue painter's tape to each square to "frame" the square. After the squares are signed, you assemble them into the quilt top. Obviously, in the photo, the quilt top was assembled first. This is not the best way to do it, in my opinion, but could work if you back the quilt top with a sufficiently smooth surface that makes writing easy.
You really want to use permanent markers made for writing on fabric. Pigma brand pens are the ones I am familiar with, and come in different sizes. For this type of quilt you probably want a fairly large pen point (nib). They come in colors too. Here is a link to some of them on Amazon:
http://www.amazon.com/Sakura-Pigma-M...dp/B00CA15JRG/
You should test any pen first, to make sure it is going to work the way you want. Some permanent markers, such as Sharpie, will bleed when used on fabric. You want a permanent fabric pen so you don't get unwanted bleeding. If you have a Dick Blick's in your area, you might want to go there and ask what is available.