I am working on an answer. As a preliminary response I would say that the notice you received is defective. Here is the statute your sister is proceeding under: § 713. Grounds where no landlord-tenant relationship exists A special proceeding may be maintained under this article after a ten-day notice to quit has been served upon the respondent in the manner prescribed in section 735, upon the following grounds: 1. The property has been sold by virtue of an execution against him or a person under whom he claims and a title under the sale has been perfected. 2. He occupies or holds the property under an agreement with the owner to occupy and cultivate it upon shares or for a share of the crops and the time fixed in the agreement for his occupancy has expired. 3. He or the person to whom he has succeeded has intruded into or squatted upon the property without the permission of the person entitled to possession and the occupancy has continued without permission or permission has been revoked and notice of the revocation given to the person to be removed. 4. The property has been sold for unpaid taxes and a tax deed has been executed and delivered to the purchaser and he or any subsequent grantee, distributee or devisee claiming title through such purchaser has complied with all provisions of law precedent to the right to possession and the time of redemption by the former owner or occupant has expired. 5. Subject to the rights and obligations set forth in section thirteen hundred five of this chapter, the property has been sold in foreclosure and either the deed delivered pursuant to such sale, or a copy of such deed, certified as provided in the civil practice law and rules, has been exhibited to him. 6. He is the tenant of a life tenant of the property, holding over and continuing in possession of the property after the termination of the estate of such life tenant without the permission of the person entitled to possession of the property upon termination of the life estate. 7. He is a licensee of the person entitled to possession of the property at the time of the license, and (a) his license has expired, or (b) his license has been revoked by the licensor, or (c) the licensor is no longer entitled to possession of the property; provided, however, that a mortgagee or vendee in possession shall not be deemed to be a licensee within the meaning of this subdivision.
I don't think you are a licensee. I think you are a tenant. A licensee is someone who has been granted occupancy from some one other than an owner.
You are a tenant at will, meaning you have an agreement with your mother and sister to live there for an unspecified period, not month to month, just indefinite as to duration. To terminate such a tenant required 30 days notice.
§ 228. Termination of tenancies at will or by sufferance, by notice
A tenancy at will or by sufferance, however created, may be terminated by a written notice of not less than thirty days given in behalf of the landlord, to the tenant, requiring him to remove from the premises; which notice must be served, either by delivering to the tenant or to a person of suitable age and discretion, residing upon the premises, or if neither the tenant nor such a person can be found, by affixing it upon a conspicuous part of the premises, where it may be conveniently read. At the expiration of thirty days after the service of such notice, the landlord may re-enter, maintain an action to recover possession, or proceed, in the manner prescribed by law, to remove the tenant, without further or other notice to quit.
Cite as Real Property Law § 228
So if your sister proceeds with an eviction suit your defense is that you are a tenant that received less than 30 days notice of termination. Your other defense is that your mother did not join in the suit which is required since she is an owner and your third defense is that if your sister is proceeding with a power of attorney from your mother and therefore your mother does not need to join in the eviction suit, that sister's use of the power of attorney is invalid since evicting you does not benefit your mother.