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Because the leaves are so thin, cuttings usually wilt before the stems develop roots; therefore, it can be somewhat difficult to try and root broken branches. However, if you want to try to propagate the broken stem, use only an inch or two of bare (green) stem with a couple small leaves, or tip cuttings, since these root much easier than the old, woody branches. Stick the cutting into a pot filled with damp potting mix and seal this inside a clear plastic bag. This will help retain moisture similar to a mini greenhouse. Place it in an area receiving bright, indirect light. In about 4-6 weeks, the cutting should be rooted and can be taken from the bag.

Here is more information: https://www.gardeningknowhow.com/ornamental/flowers/poinsettia/broken-poinsettia-stems.htm

Yes, this shrub-like plant has roots and can last months or years given the proper growing conditions. For more information on growing these plants, these articles should help: https://www.gardeningknowhow.com/ornamental/flowers/poinsettia/poinsettia-care-how-do-you-take-care-of-poinsettias.htm

https://www.gardeningknowhow.com/ornamental/flowers/poinsettia/how-to-make-poinsettia-turn-red-make-a-poinsettia-rebloom.htm

Reblooming a poinsettia can prove difficult, especially outdoors. Poinsettias require 14 hours of uninterrupted darkness every night during the month of October, when flower buds are developing. Any light exposure during any one night can result in a day's delay in your poinsettia's red color. Protect your poinsettia from light starting October 1 to attempt a rebloom. Since it's already December, your plant may not turn at all. That said, you could TRY covering the poinsettia plant each night (with a box turned over top of it - make sure it's large enough) to block out any light. Do this for the remainder of the month and keep checking to see if this helps any with the color.

Perhaps they have been too-well watered. If you were to water them every day, for instance, they would get root rot and start to brown, shrivel, and probably die. Do not water again if the soil has water dripping out if you pinch some of the soil between your fingers. Wait until a pinch of soil will stick together when you stop pinching, but fall apart when you touch it. This article has more information: https://www.gardeningknowhow.com/ornamental/flowers/poinsettia/poinsettia-care-how-do-you-take-care-of-poinsettias.htm

What are generally regarded as flowers on poinsettia are actually leaves. Their flower buds are very small.

Regardless of whether you meant flower buds or the distinctive red leaves, there are many reasons they can fall and all are related to stress. This time of year, I would suspect the likely culprits are either the plant is underwatered or is being exposed to a mild chilly draft.

The red leaves are maturing and assuming your plant originally was a red color it is entering another growing cycle.
Here is a link about the care of poinsettia.

https://www.gardeningknowhow.com/ornamental/flowers/poinsettia/poinsettia-care-how-do-you-take-care-of-poinsettias.htm