Hello! So sorry about your hen!
I know this is going to be a lot of questions...
Could you post a pic of the abnormal droppings? (are they runny and/or infrequent?)
Could you also get a pic of her side-profile so we can see how she carries herself
(certain postures can suggest/indicate reproductive issues).
Is she frequently sitting or carrying herself low to the ground?
How long has she been like this?
Is she eating/drinking?
Do you know when the last time she laid was?
What does in between her legs (below the vent) feel like (swollen, hard, squishy or normal)?
Also, how old is she? She looks like she has been over mated.... do you have one or more cockerels? She also looks twisted in that photo. Does she have wry tail? Is she able to walk? She looks like she has her wings out to balance herself or is that the photo angle?
The white poop is usually urates which are the bird equivalent of urine. If nothing else is being passed then it can indicate that her gut is obstructed. Being egg bound can do that, but there are other possible causes. Having answers to the above questions and more photos or better still a video (needs to be downloaded to You Tube or similar and a link to it posted here) will help to narrow down the possible diagnoses.
I'm so sorry. At least she is no longer suffering.
I don't know where in the world you are but in some states in the USA it is possible to get a necropsy done at a subsidised rate.... in CA I believe it costs £20 which is well worth doing because there may be things that you can learn from her death that might benefit the rest of your flock.... things like excessive fatty deposits or Fatty Liver Haemorrhagic Syndrome would suggest a dietary imbalance. High levels of internal parasites would indicate that it would be wise to worm the rest of your flock etc. If you are of an enquiring mind and not too squeamish, you could also cut her open yourself if a professional necropsy isn't an option. It is surprising how much you can learn even without medical training. I appreciate that not everyone can get their head around cutting open a loved pet but some people can with a little encouragement.
The video linked below is of a lady doing a DIY necropsy on a hen that had salpingitis.... the video says egg yolk peritonitis but I believe infection of the oviduct (salpingitis) which is the large mass she cuts into was the primary cause.
Warning .... the video is very graphic so don't view it if you are sensitive...
Tried to link it but a graphic image came up with the link and didn't want to leave it on view just in case... If you are interested in seeing it, ask and I'll post it.