I live in NH (no income tax) My employer transferred me to MA about 5 years ago. At the time of transfer I was told they would not have to deduct taxes or they would compensate me for the difference. They never deducted anything. Now I believe I am going to be fired and will need to collect unemployment. Will I be liable for back taxes? What are the penalties for the employer?

If I understand correctly - you have worked in MA for 5 years, but your employer never deducted MA income tax from your paycheck, and I would also guess that they have not been deducting MA unemployment insurance either, correct? If/when you are fired, if you file for unemployment benefits from MA as part of that process they will contact your employer to verify the data you submit about your job, including verifying that you lost your job through no fault of your own. If your employer has been voilating MA tax laws by not paying unemployment insurance for you, I suspect that the MA authorities will have not take kindly to that. Accordintg to the MA website possible enforcememt tools they could use against your employer include:

Ongoing, random audits of employer accounts to see if all workers and wages were properly reported.
Late filing penalties of 10 percent of the contribution due, with a floor of $35 and a ceiling of $1,000 for each late quarter.
Levies on both a delinquent employer's bank account and state or local government funds owed to the employer after a court judgment and formal notice.
Liens on real estate.
Assistance from the state Department of Revenue in locating employers not registered with DUA.
Civil complaints brought by DUA attorneys.
Suspension of a delinquent employer's liquor license after a court judgment and a hearing.
After notification, public disclosure through the publication of a list of delinquent employers owing more than $5,000.
Prosecution of individuals and principals in corporations by the attorney general's office for failure to file or pay unemployment insurance contributions.
A minimum $2,500 fine, up to $10,000 maximum, for filing but failing to pay unemployment insurance contributions.
A minimum $10,000 fine, up to $50,000 maximum, for each quarter and/or a state prison sentence for a felony conviction on contribution evasion charges.

As for your back taxes - if you have not been paying any MA tax at all, you owe back taxes plus interest plus penalties for failing to file and pay on time.