Yes you can! Undeclared voters — those not registered with any party — can vote in either party primary. In 2009, partisans unsuccessfully tried to make changes to this rule. (Also see IndependentVoting.org for more info and connections for independents in New Hampshire)
From Steve Peoples at CQ Politics, Ayotte's Lead Narrows in N.H. GOP Senate Primary:
Ayotte led 37 percent to Lamontagne's 30 percent among likely Republican primary voters. Deep-pocketed businessman Bill Binnie garnered just 13 percent, despite having spent nearly $6 million out of his own pocket, while another conservative favorite, Jim Bender, finished last with 12 percent.
The survey of 1,134 likely Republican primary voters (New Hampshire's open primary system allows voters of all political affiliations to participate in the contest) was conducted Saturday and Sunday. The margin of error was 2.9 points.