The former Northern Ireland international centre-half nodded a header from Guy Butters off the line midway through the first half.
Brighton thought they had taken the lead a minute later when Chris Iwelumo headed in a cross from Nathan Jones, but referee Lee Cable ruled it out for a push by Iwelumo on Hunter.
The pattern of Brighton attacking continued in the second half, but they struggled to create chances.
Seagulls manager Mark McGhee made a triple substitution with 12 minutes left, including the withdrawal of 25-goal Leon Knight, in a vain attempt to break the stalemate.
McGhee said: "Even though I made the substitutions I was not convinced it would make any difference. There was no change to the shape we were going to make which would change the pattern of the game.
"Rushden came with a strategy to sit in, crowd the box and defend deep and we could not break them down.
"You could understand the way they played in their position. In a way it was negative and it was a difficult afternoon for us.
"We have still improved our play-off position. We are now four points clear of the team in seventh."Hunter was delighted with the result. "It is nice to get a point and a clean sheet," he said. "We achieved what we came for.
"I think we have done okay in the last few weeks and had nothing to show for it."