Firearms licensing (with respect to the License to Carry Firearms) in Massachusetts utilizes a two-prong test, and grants significant discretion to the chief law enforcement officer as to whether the license will issue. Although the offence will not disqualify you as a matter of law under s. 131, the existence of the record and the underlying incident may be considered by the chief as evidence of "unsuitability", which could result in the denial of an LTC/A.
Should the license be denied, you may be able to appeal the denial within 90 days to the local district court. (M.G.L. c. 140 s. 131).
Additionally, depending on the age of the criminal record, and the circumstances of the case, you could consider applying for a Massachusetts Firearm Identification Card. The FID will only permit you to posses non-high capacity rifles and shotguns (no handguns), but such license must be issued by the chief law enforcement officer if the applicant is not statutorily disqualified from obtaining said FID. (See M.G.L. c. 140 s. 129B).