Hospital is recommending a short-term stay in a skilled nursing facility for Mom. Does anyone know whether this type of facility will require her social security or VA to pay for it?

Short term rehab in a Skilled Nursing Facility is paid by Medicare and has no impact on SSI or VA payments. You can check with your mother's Medicare coverage plan to make sure it's covered, but I'd be surprised if it wasn't. The short term rehab stay is normally 20 days and then she'll be released back home.

If the Skilled Nursing Facility wants mom to stay there permanently, then she'd have to apply for Medicaid to cover the expense if she can't afford to pay privately. The decision would be up to her and her POA, however. Then if Medicaid picks up the cost, it would affect her SSI and VA benefits which would be used to help pay for her SNF costs; she would receive a small monthly stipend to cover personal costs only.

You need to find out if Medicare will be paying for this. If so they pay 100% for the first 20 days, 50% the 21 to 100days. The 50% not cover either Mom pays or her secondary pays for partial, all or none. The only time SS gets involved is if she goes beyond the 100 days and needs Medicaid to pay for her care. Then they will ask for her SS. If VA means Aid and Attendance, you can't have it and Medicaid.

Usually when u go over 100 days its found that a person needs 24/7 care. At that time a decision needs to be made if the person goes to LTC, an AL, care is set up in the home, or live with family.

Speak with your Social Worker in Hospital and she can research this for your area. In general SNF is covered by medicare for a certain number of days if ordered by the doctor for skilled nursing issues. That however doesn't really include degenerative spine issues that I know of. That would perhaps be more rehab with finding ways of movement to increase mobility without pain.
I would discuss with the Social Service or with the SNF upon admission.
Long term care isn't usually done in either SNF or in Rehab.
Wishing you good luck.