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I agree, keep it whole. Ask your cousins if any of them would be interesting in "hosting" the quilt at their house for a certain period of time. Kind of like a library loan. It would always come back to you and you could send it out for loan to the next person. Don't forget to put yourself in the loan schedule. Some probably wouldn't want to participate, some would. I like the idea also of taking photos of those squares that go with certain people. Maybe print the photo on printable fabric and make a wall hanging or pillows.
Luv Quilts and Cats
Never underestimate the healing effects of beauty. - Florence Nightingale
I would not take it apart, but that is just my opinion. It is your quilt and you have the option to do as you please. If it were me, I would be honored to have a family heirloom. Not everyone likes quilts and you receiving this quilt may have been because they knew you would take care of it more than any other family member. I would hate for you to deconstruct it if another member is against it. Now if the quilt is worn or tattered, then deconstruction could be warranted.
Keep it whole! You could possibly pass it around for a certain amount of time to those who would cherish and care for it while it was their turn to have it. Otherwise, I would just take a photo of their block and copy it onto fabric. Then either make a pillow or small wall hanging with it.
I think I would have to agree -- don't deconstruct the quilt. My grandmother made LOTS and LOTS of quilts which were shared with her two children and her grandchildren (I think about 15-20 of them). A lot of these were gifted directly such as to the grandchildren, but a lot of them had been retained by my grandmother. I am sure there was some division of those quilts, but I wasn't physically able to be at my grandmother's funeral so I really don't know. I am relatively sure that most of them remained with my mother, since my grandmother was living at my parents' house for many years. When my parents passed away in late 1998 and 1999, the quilts were divided equally between my parents' children. I wouldn't even consider deconstructing any of the quilts -- they were all in extremely good condition. No one complained, so I think that division was fair. One of the quilts (a king size top) was a grandmother's flower garden quilt. I was not doing quilts at that time, so I sent it to an Aunt on my DH's side and had it hand-quilted. It is very precious to me because my grandmother made it.
On the other side of the family, my husband's aunt (she's 85) took a serious fall, breaking her hip. She had to have surgery twice and dimentia had struck. She was no longer mentally competent to maintain a household, so she gave things away. Some of the things she had not specifically gifted included quilts, some complete and some completed except for binding (this one, in particular, had hen-scratch embroidery in beautiful shades of blue and blue-checked gingham) Even before I got the quilting "bug" I would never have thrown that away, but my SIL said, "Oh, just throw those away." I said, "Not on your Life." I have them today! So, some family members treasure hand-made things, some don't. Don't deconstruct the quilt unless it's deteriorated!