The American right is quite diverse, but there are reasons why people of various philosophical and political bents wouldn't mind labeling their opponents progressives.
This group is made up of people who love the American founding and the ideas of life, liberty and property that shaped early American political thought. For many years the American left was known as liberals. To have classical liberalism associated with big government paternalism in the modern left really aggravates these people. For Example:
Here I make a plea, addressed to conservatives and libertarians,
regarding the word liberal: please do not describe leftists,
progressives, social democrats, or Democrats as “liberal.” I do not
ask that you describe yourself as “liberal.” Continue to call yourself
“conservative” or “libertarian.” I propose only a single step: don’t
call leftists “liberal.” By this single step, we can make great
Many on the right want to take back the term liberal and use it to describe the negative rights loving liberals of the 1700s. They think liberal is a misnomer when it comes to today's left. But then what should they call modern Democrats? Progressive is a term already used. Silly notions of "the right side of history" and progress are much less a concern to classical liberals than taking back the word liberal from the left.
Radical libertarians actually do care about progress. They want to change the world into one that is free of the coercive hand of government. They, of anyone on the American right should care about this misuse of the term "progressive." Actually, not so much. The term progressive in American politics is a loaded one. It dates back to the progressive movement in the early 20th century. However only a few sentences into the Wikipedia article on the progressive era we run into the line:
Many progressives supported prohibition of alcoholic beverages,
ostensibly to destroy the political power of local bosses based in
saloons, but others out of a religious motivation.
This line taints the progressive movement's legacy irrevocably for radical libertarians. In their eyes, prohibition is a classic failure of government action; it's the worst kind of meddling in non-violent free action. Radical libertarians would love to label their opponents as progressive. The failures (in their eyes) of the progressive movement in the 1920s are exactly what radical libertarians would love people to think about when modern progressives ban straws to try to tax soda.
Finally, philosophical conservatives have no problem with labeling their opponents as progressive, because they fundamentally believe that it is better to conserve than to progress. Conservatives are quick to look at the failures of utopian revolutions as the result of an obsession of progress instead of an emphasis on conserving the good with prudent reforms. Michael Oakeshott's definition puts it succinctly:
To be conservative, then, is to prefer the familiar to the unknown,to
prefer the tried to the untried, fact to mystery, the actual to the
possible, the limited to the unbounded, the near to the distant, the
sufficient to the superabundant, the convenient to the perfect,
present laughter to Utopian bliss.
A conservative doesn't hate progress, but to him, an overemphasis on progress is unhealthy and dangerous. Conservatives are happy to let their political opponents be defined by an obsession with progress. They often point to movements obsessed with progress like France in 1789 or Russia in 1918 that quickly turned to poverty and bloodshed.
This answer doesn't, however, explain why populist nationalists like Donald Trump would be okay with labeling their opponents as progressive. Anecdotaly, I have noticed that Trump seems to use the term liberal more than progressive to describe his enemies on the left.