Little Lulu

I have cut all my shirts (all 100% cotton - good quality) and am beginning to piece my baby quilt for my soon to be great grandson who will have his name. It just dawned on me that I need to know the following before I proceed so please help (I hope this will be my last question- Ha).

It is obvious that my late husband's shirts have been washed, dried and ironed numerous times. I am using a Kona white cotton 10" square pack as the background for my half square triangles. I don't usually pre-wash my fabric so should I wash the Kona white squares, binding and backing or will it be ok not to wash, dry and iron the background, binding and backing materials?

Thanks again for your help. Cutting my husband's shirts has been an emotional roller coaster, but I really want to do this for the family. Again, thanks.