A landlord can only evict you if they go through the proper legal process, regardless of your agreement. There are certain situations whereby a landlord can obtain an eviction with little or no notice to the tenant, but those situations typically involve criminal acts being perpetrated in the apartment by the tenant, which doesn't seem to be the case here.
As such, a landlord must give you at least a 7 day Notice to Quit (which is also only available in certain limited circumstances), and will likely have to provide you with a either a 14 or 30 day Notice to Quit. The Notice to Quit is a document that must be served upon you informing you of the date on which your tenancy will be terminated. The landlord cannot even begin an eviction action in court until the period given on that Notice to Quit has expired.
All this is to say that a landlord, if a tenancy has been established, cannot simply give you a date and tell you that you have to be out by that time, they have to follow proper legal procedure. Massachusetts, as a general matter, is one of the states most protective of tenants, and landlords who engage in self help (e.g. changing the locks, moving you out on their own, or otherwise not following proper legal procedure) are typically punished harshly by the courts.
If your landlord has not followed the steps required to effect a legal eviction, then you do not have to leave the apartment. Even if they have been able to secure a judgment for eviction through the courts, they still have to go through the extra step of getting a court to approve an execution on that judgment (the court has to give the OK for them to actually move you out). If the landlord has recently gotten a judgment for eviction, you may also be within the time period during which you can appeal that judgment. Finally, even if you do not appeal the judgment, you may be able to file for a temporary restraining order to prevent the landlord from executing on the judgment.
This can all be very confusing for someone facing an eviction for the first time, and there's simply not enough room here to explain all of the various aspects of eviction law. Moreover, given the few facts given I cannot make any determinations as to whether a tenancy has been created, whether proper eviction procedure has been followed, or what, precisely, your rights are in this situation. As such, I highly recommend you visit the website www.MassLegalHelp.org for further information, and I would further suggest that you contact a local legal services agency for specific advice and or/representation. I wish you the best of luck.
If you can't move, simply don't. Wait until you receive a notice to quit from your Landlord and go from there. Your landlord can't physically remove you. It sounds like you have an established tenancy.
The response given is not intended to create, nor does it create an attorney-client relationship or an ongoing duty to respond to questions. This is not intended to be anything other than the educated opinion of the author. It should not be relied upon as legal advice. The response given is based upon the limited facts provided by the person asking the question. To the extent additional or different facts exist, the response might possibly change. For a specific answer to your question, you should speak with an attorney who practices in this specific area of law.