Responding to Arjen Dijksman, Bohm and Bell must have chosen a singlet for the initial state because it is the simplest case. There are then just two possible ways to assign the spin components of the spin 1/2 particles such that they cancel along some common axis. The antisymmetric superposition of these two primitive assignments is the one that forms a singlet. It has no free parameters, so Bell's analysis must be mixing in the symmetric superposition, which belongs to a triplet thus violating conservation of angular momentum. By the way, I just became aware that C.S. Unnikrishnan developed this argument [1] in some detail 5 years ago.
You can prove that half-odd-integer spin states change sign under a roation of 360 degrees by considering the eigenstates of the spin component along the axis of the rotation. These components range from -n/2 to +n/2 for spin n, so if n is odd, n/2 times 360 becomes an integer times 360 plus or minus 180. Since the basis states all change sign, it follws that any other state, being expressible as a linear combination of the 2n+1 eigenstates, also changes sign.
Thanks also to Arjen Dijksman for the Wheeler reference.
1. C. S. Unnikrishnan, Correlation functions, Bell's inequalities and the fundamental conservation laws, http://arxiv.org/abs/quant-ph/0407041v1