It's a common feeling in the poly community that if your primary relationship is not strong, you have no business forming other relationships with other people. In this case, it sounds as though HER relationship with her BF is not strong.
As for moving in, I've never seen this be a good solution to problems. Like you said, she's a grown-up and it's her responsibility to solve her own life's issues. By all means, help her, support her, be a shoulder for her to cry on or whatever, but I wouldn't dream of opening up your house to that kind of toxicity.
Someone already said it, but I'll repeat for emphasis: She's lying to her boyfriend about you, and she's lying to you about lying to him. Who knows what else she's lying about? Moving in with her means you'd be opening yourself up to the consequences of that dishonesty.
If it does come to a situation where she absolutely needs somewhere to go and has nowhere to turn to but your place, make sure you put a very definite time limit on it for her to get her life together and get out on her own. Before she packs her bags, help her develop an action plan in terms of getting a job and finding an apartment if that's what she needs.
“As I am sure any cat owner will be able to tell you,
someone else putting you in a box is entirely different
from getting into a box yourself.” —bisexualbaker