Social media gag order?

I doubt very much that you can remove the free speech rights of anyone anywhere from saying pretty much anything excepting "fire" in a crowded theater. Now it is about protecting Mom and her assets. Who is the POA? Who has guardianship if there is any? Who is in charge of safekeeping, because valuable items need to be kept safe, whether in storage or some other way, by the POA. Good luck. Hope you will update. Meanwhile, the answer to to stop partaking of Sibling's rants, and to tell people who send screen shots, et al, to please refrain, that you do not have time nor heart for this.

I’ve not heard of a gag order on a matter like this. I would take screenshots of what’s being posted and take them to your lawyer for advice. Seems like if it’s firm that your mother won’t be returning home then getting her property and home sold to help pay for her care would be the quickest way to end this

Ask your attorney about writing and serving the sibling with a cease and desist order.    I don't know what the penalties for breach are w/o researching your state's statutes, but your attorney could tell you more quickly.   She/he can cite the appropriate statutes.  

Alert the local police and ask if they can do periodic drive-bys.   Some may have the time; others may not.   

I don't know whether you yourself have access to the house, given the situation with the sibling, but if you do:

Find a good security service and get a system installed as quickly as you can.   Also, alert the neighbors to be on the look out.    This won't help when they're asleep at night though.

Get those little window alarms and put them on the inside of the windows.   Put chairs and other furniture in pathways to block anyone's easy ability to walk through the house.   

There are probably other good methods, but the issue turns basically on the irresponsible actions of the sibling.   You may end up needing to request something more powerful than a cease and desist, something like a restraining order, although I don't know whether or not a court would be sympathetic, especially given the apparently discord in the family.

You can also block windows and doors with furniture.

All of this is moot thought if your sibling is denying access to the house.   In that case, you'll have to rely on legal action, which you should discuss with your attorney.  I would consider a PPO first though, if your attorney thinks you can get one that prevents the sibling from sharing data.    I don't see why a "gag order" doesn't apply though.