I have had a look at a few reviews of this book, and they all describe it as 'extreme' or 'hard core' calisthenics. Given that you have HMS, and that you describe yourself as very out of shape, it may well be too much for your body to cope with at the moment. However, calistesthenics does not have to be as extreme, and it may be that some of the beginner type exercises could be a good starting point.
One of the problems with exercising when you have HMS/EDS-H is that our bodies often do not behave the way they are meant to. It is common for incorrect muscles to be used for movements, for joints to hyper-extend, and for us not to find it hard to to the exercises right due to poor proprioception. We often need advice and guidance from a physio or exercise tutor who understnds HMS to ensure that we use the right muscles, do not extend our joints into the hypermobile range and to do the exercises correctly.
Diagnosed HEDS December 1st 2005. DD1 (20) HEDS and scoliosis (now corrected by surgery), diagnosed June 2006. DD2 (18) mild HMS. Son (11) some hypermobile joints, poor muscle strength and seems to be developing scoliosis as well, woopee!