Personally, I would use come off and not come out.
come out does mean that something is being removed, but from INSIDE something else. A similar meaning to extract (not a synonym):
This screw will not come out
I am trying to extract the screw from its hole, but I cannot (because it is too tight maybe?).
come off means something is ON something else and I want to remove it.
The stains from the meat sauce won't come off.
I am trying to remove the stains, but I cannot (because they are too thick?).
Now with this modification, on to your sentences:
a. The stains from the meat sauce won't come off.
b. The stains from the meat sauce won't be removed.
b to me sounds like the act of removing the stains will not happen. That is, nobody will do it or even try it. a means that no matter how hard you're trying, you cannot manage to remove the stains.