There are several possible reasons for the development of side stems which are so weak and thin that the inflorescences cannot be kept erect without staking. The following is more information.
Research has found that cutting quality plays a part in stem breakage. Larger sized cuttings do not tend to break as easily as smaller, weaker cuttings. Early in the production cycle the plants should also be spaced closer together so that the plants will grow more upright. Plants grown with a wider spacing are more likely to produce lateral shoots that can grow out horizontal over the edge of the pot. Support rings will help support the plant and prevent stem breakage, but the cost of the rings and the labor to install them has to be economically justified by the grower. Calcium deficiency also has been reported by some researchers as a cause of weak stems, as calcium is an important constituent of plant cell walls. Varieties can also differ in stem strength with 'Success' and 'Red Splendor' being varieties which are more resistant to breakage. Some of the free-branching varieties produce so many shoots that crowding and reduced light intensity occur. The number of shots can be controlled to a certain extent by limiting the number of nodes below the pinch to 5 or 6 or by the removal of some of the lateral shoots. The first method is practice more often, since pruning can be an expensive operation. Limiting the percentage of ammonical-nitrogen being applied may also help avoid stem breakage. Ammonical-nitrogen promotes vegetative growth that may lead to weaker stem. Since thicker, stronger stems generally result for growth regulator treatments, some growers apply growth regulators to improve stem strength as much as to control height.
Information from the Poinsettia Problem Diagnostic Key on Physiological Disorders from North Carolina State University http://www.ces.ncsu.edu/depts/hort/poinsettia/corrective/a11.html