SALT LAKE CITY — From the moment the Affordable Care Act passed in 2010, most elected officials in this sturdily Republican state have been eager to squash it.

But something surprising is happening here. Despite deep uncertainty about the law’s future, Utah recorded one of the biggest increases of any state in residents who signed up for coverage under the act this year. Now, the state is seeing a surprising burst of activism against repealing the law — including from Republicans.

“I’m naturally a really quiet person, but if I sit and do nothing and they take it away, how can I live with that?” asked Kim Nelson, 54, a Republican second-grade teacher who buys coverage through the Affordable Care Act marketplace and was recently treated for breast cancer. She has been calling and writing her congresswoman.

Many Utah residents still detest the law, but the activity here, and in other politically conservative states, highlights the challenges Republicans and the White House face as they struggle to come up with a palatable replacement.