If someone's spouse is in a NH can he/she claim the spouse on their income tax?

Of course he can, and he can deduct the cost of her care. Have him see a tax consultant who can help him maximize his return.

Why not, they are still married.

Yes, but if the other partner is incapacitated an no longer able to handle their personal and financial affairs, the person handling the filing needs to obtain representative authority to file a MFJ return and sign for that spouse. Make sure your court ordered guardianship, POA, etc states specifically "for all federal, state, and local tax matters"
otherwise, there can be problems accessing a spouse's tax info for separate matters on MFS returns, business debt, or anything that was specifically the spouse's private information. Being a spouse is not an automatic account access, even in a community property state, and definitely nor with the IRS

We're talking about MFJ not married filing separately. The last poster sort of overcomplicated the issue. It is true of course that the IRS protects the separate info of every taxpayer.