It depends. Two separate questions. As far as the Health Insurance is concerned, this is something that you would have to research on with the state of Massachusetts to see if you qualify to obtain health insurance there as a nonresident of Massachusetts. In truth, however, the answer to that question does not directly affect your state tax situation.
Since you are an Alaska resident, who works in Alaska, you would have no requirement to file a Massachusetts return unless you work in Massachusetts as a nonresident. Simply having MA health insurance will not change in and of itself your residency.
Things get tricky, however, if you must establish MA residency to get insurance (which you decide to do), but maintain your AK home and job and continue to work exclusively in AK. The reason is because MA would claim you are a resident and therefore must pay income tax to MA on all of your income. If this is the way to get the health insurance and care you desire, you would want to think if paying state income tax that you would not have had to pay otherwise is worth that price.
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