If you hold promotional poker tournaments where the participants do
not pay to play, and then you give a prize to the winner, is that
considered "illegal gambling" in either NH or MA?

If the prize is not money then there is definately no gambling taking
place as gambling is defined as playing for money:
However if you all put together and buy the prize and then play for
it, that is gambling.
Gambling is also defined as 'a transaction depending on chance' and if
they do not have a stake in the prize then it is not gambling.