There was no single "Minute Company."There were several, each comprised of men from their own County Militia.
Determine what County Militia your man was a member of, and start with records of that County.
All able-bodied free white males aged roughly 16-60 (age range varies by time and place) were considered part of the County Militia as a matter of law.There were no special 'enlistment' records.Surviving militia muster rolls were made when men reported for regular drill and nose-counting, and unless otherwise stated do not signify active-duty service.
County Militias were intended to defend within the County, but on relatively rare occasions were sent elsewhere.Some of the MD Minute Companies were even sent outside MD.But usually a militiaman's 'active duty' service would be on the order of patrolling, guarding prisoners, assisting to escort supplies bound elsewhere, or sometimes to staff a stockade or blockhouse that might be called a 'fort'.
You will want to look at estate, land, tax and Court records for the county your man lived in, to see if there is a connection with ancestral family.