MANCHESTER, N.H.—Gary Brockney isn’t happy that Republican Sen. Kelly Ayotte announced just over a week ago that she could no longer support her party’s presidential nominee, Donald Trump.

“It’s just ludicrous. If you’re a Republican, you unite—you don’t flip-flop like she’s been doing,” said Mr. Brockney, a barber in Wolfeboro and enthusiastic Trump supporter.

In...