Hi Cybil101, welcome. It's nice that you're trying to care for your friend, and you reached out here. You did the right thing, because this is a supportive place, full of kindness, love, etc. No judgement whatsoever from anybody here. You're also doing the right thing supporting your friend.
I'm sorry they lash out at you and say the things you mentioned. It could just be the fact they have trust issues and are afraid of judgement, rejection, failure, getting too close, and so forth. Perhaps they have "walls" up so to speak and like a bubble, they don't want anyone coming inside. That's common for those with mental illnesses.
They're not a burden, and are worthy of love, yours included. But once again, it's common for those who are mentally ill to feel that way and say those things also.
I'm sure deep down they appreciate your kindness and efforts in supporting them, they may just be confused and having a tough time and they're unsure of how to show their appreciation. But I'm sure they are thankful for you.
Perhaps they think that because their family is unfortunately unsupportive (thats so sad, I'm so sorry), that others like you may be too. Of course you're the opposite, you ARE supportive, but they probably figure that just because their family is that everyone may be like that, which isn't true.
I hope this answers your questions a bit, I'm sorry if it doesn't, these are just my personal thoughts and from my personal experiences, plus you got some helpful answers above.
Please know that you're doing everything you can, and just being there is enough whether that means talking to them, listening, giving them space, whatever. I'm sure your friend does appreciate it and is grateful, even if they're struggling and not sure what to do or say.
I do hope that they can get the therapy they need and deserve, and they can work on a road to recovery.
Best of luck to you and your friend, I hope your friendship is okay, and nothing happens to it or either of you.
We're here for you, and your friend, even if they're not on the BB forums, always.
Please stay safe, and your friend too, take care.