My mom wants to come home from an ALF. Can she sign herself out?

Time to talk to her facility. They will know if she exhibits competency to the extent that she is able to act in her own behalf.
Next step is diagnosis. If she is diagnosed as incompetent then she must stay where she is; if you need guardianship or conservatorship to accomplish that then that will be step three.
And yes, of course you can refuse to let her live in your home, and I will tell right now that if you allow her into your home it is unlikely you will get her ever to leave it again, as you seem very unable to level with her honestly about what is now happening in her life and what your limitations are.
I am so sorry. This is very tough. But you cannot tippy toe around it without a disaster happening that you cannot get out of.
It is time to visit her ALF. Time to arrange her exams. Time to level with her (and you can do this often with the social worker at the facility right there) that she cannot be safe on her own any more. That the first step to examine all of that is to have her tested, and "see how you do". That for now she is where she is. That coming to you is not an option; that you are sorry but your human limitations mean you are not capable of that. Time for honesty. While that hurt, this also hurts, but this is going to hurt a lot MORE without leveling with honesty.

If this is a new behavior it may be she is Sundowning. The home she wants to go to is probably not yours but the one of her youth, in her long-term memory. My MIL is in LTC with mild dementia and short-term memory loss. At the quarterly care meetings she always asks when she can "get out of this place" (even though she was there when her house went into foreclosure and is no longer hers) we tell her she can leave when she can do her ADLs by herself and prove to her doctor that she is able to live on her own. She doesn't "like" this answer but she can't argue against it and it relieves us from being the "bad guys".

When she brings it up divert her to a different topic or give her the excuse I just provided and see what she does with it.

Yes you can and SHOULD and MUST refuse to let her live with you.

Buy better locks, get some guard dogs, dig a moat, build guard towers and and hire some archers.... just don't let her in you house, or you'll have an awful time removing her. If she shows up with her suitcase on the doorstep, just don't answer.

If your home, you decide who lives there.

If Mom's name's on title, she would be Landlord & you Tennant. Landlords cannot just move back in whenever.

This is more about FEELINGS I think, than actually moving house.

Feeling lonely is likely. Chat to the AL Manager about getting Mom engaged in some activities. Memory problems effect refuced socialising. That's where arranged activites come in. (Social engineered contact if you want to be fancy). Think play dates for elders. She may refuse.. but with some gentle positive coaxing may join in & be less lonely.

Going 'home' can also mean 'I want to be young again' or 'be happier'. If so, reminiscing over fond memories & treasured photos can help. Again, this may need someone to share.. hiring a companion aide to go out for coffee once a week is another idea.

So I would avoid any talk of her moving in with you, stop that guilt train at the station.
Focus on assisting Mom to connect & find *home* where she is.

Hi,
If you're her POA then you agreed to act in her best interests if she's not capable of doing so. No-one's expecting you to have her live with you, but as POA you should make sure she''s safe and that her finances are taken care of. If she has short term memory loss and is unable to pay her bills or remember to eat then it seems like the POA would be active. She isn't capable of making good decisions. You will need to get her to her physician and have her assessed for dementia and get the POA activated.
These are helpful:
http://www.dementiacarestrategies.com/12_pt_Understanding_the_Dementia_Experience.pdf
https://tamcummings.com/stages-of-dementia/

The Alzheimer forums are helpful as well...https://www.alzconnected.org/discussion.aspx

You do not have to let her live with you. AL isn't prison, she can leave anytime she wants. The real question is, can she actually make the arraignments to move by herself. If not, she's not going anywhere. You aren't helping her to leave and nobody else should help with this.

If she doesn't have dementia, you can tell her that you will be withdrawing as her POA if she decides to ignore your counsel and head out on her own.

If she has dementia, maybe just ignoring the comments will be enough, you might try not answering her calls for a few days or weeks to see if she can actually set things up on her own. Stay in touch with the AL to make sure she is doing okay.

Can she understand being told that she needs to stay there until she gets stronger, the doctor says or some other point that would get her to give it more time?

If this is new behavior, I would request they check for UTI, it could create this type of discontentment.

Can you provide more information about her condition? It helps with answering your post.

You can always refuse to let into your home.