The Spitfires - so strong in the FA Cup over the past few years - are out at the first stage following the Bulls' headline making performance.
It has added to Eastleigh's problems of late, with the injury hit Spitfires out of form in the Vanarama National League.
But the boss won't throw in the towel and retains confidence that results will come.
"Credit to Hereford, they've done great, and credit to their fans. They must have brought about 1,000 supporters and they've had a big influence," Hill said.
"This club has had a lot of joy in the FA Cup - to the point where it's become expected.
"I'd imagine it will be rife, calling for Richard Hill's head, and I get that. But I'm not going to walk away.
"What can we do?" the manager told the Daily Echo. "We either sulk or we get back to work and find a way to win a game of football."
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