My BPD mother becomes angry and verbally abusive if someone is not feeling well or hurt for some reason. I have not seen her show empathy.
Yes, it is puzzling because lack of empathy would be no response, but why the increase in abusive behavior?
My own explanation is that BPD mother is in Victim position on the Karpman triangle. Everyone else is either Persecutor or Rescuer. If you are not being attentive to her needs, then you are Persecutor. She doesn't see her behavior as being abusive- she sees it as - she's hurt and so she hurts back.
If someone else is feeling poorly or hurt- there's a natural shift in attention to their own situation. They, themselves, are in a position of need- whether it's care, or understanding, or empathy. That doesn't fit into her thoughts scheme. They are not in Rescuer- they must be in Persecutor and doing this on purpose.
Another possible explanation is that she may use illness as a means of getting her needs met. It's hard to know if what is going on is real or not as she sometimes exaggerates it, or feigns it. If she knows she does that, then she assumes other people are doing that too and so gets angry.
Whatever the reason, it helps to reframe this into being a result of their thinking rather than it be personal. It's not about anyone else. It's how they process these situations.
I think both of these dynamics could play a role at the same time, or otherwise explain why they behave this way when confronted with others' needs.
In my own experience, BPDxw added a "twist" on what others here have described, in that she'd initially be very concerned if I was sick or hurt, and would urge me to go to the doctor if I was reluctant, and get rest/follow the doctors' orders. This lasted about a day. But would come up again. her initial concern was always brought up again as "proof" that she was a caring empathetic person who I didn't deserve or couldn't match in empathy. I'd get comments about how she told me to go to the doctor & that was proof she cared more for me than I even cared about myself. Whatever... for a day at least, assuming her concern was ever genuine in the first place.
when I was "out of commission," i.e. wasn't able to help her watch our daughter, put her to bed, help with housework in the evening, etc., BPDxw's concern would quickly evaporate into impatient demands and insults.
On one occasion, after an injury the doctors told me to take extra bed rest; they said take another day off after the pain stops, to make sure I didn't reinjure my muscle or make it worse. Day 1, BPDxw was normal, but by Day 2 she was making jabs about my toughness and asking how long I "planned" to be hurt for, as though I was faking it all.
Needless to say, when BPDxw was hurt or sick, she expected a lot more than she ever provided. I remember when she was ordered bed rest for the last two weeks of pregnancy, my mom flew across the country to stay with us and wait on her. She related a story that she heard my mom on the phone talking to my brother and laughing about something, while she - BPDxw - was in bed, and "could never forgive the callousness my mom displayed" ... I guess everyone was supposed to mope around silently while she laid in bed, ate ice cream and watched reality TV shows...