A now-deleted comment asked if perhaps the person in question was thinking of private turnpike roads, such as the Philadelphia and Lancaster Turnpike in Pennsylvania (which was the first major paved and macadamized road in the United States).

If that is what the person is referring to, then they would be not quite right in thinking that such turnpikes were made by private individuals banding together without state involvement.

While turnpikes like the Lancaster turnpike were corporations that raised funds from investors, they were established with very active state government support. The route (and potential benefits) of the Lancaster turnpike were analyzed by a Select Committee from the state legislature in 1791, and the legislature passed a bill authorizing Pennsylvania's governor to establish a turnpike Corporation in 1792 (corporations at this time needed explicit legislation for their establishment). Supplemental legislation in 1795 even regulated the maximum width of the road allowable for the turnpike, and prohibited tolls from being collected on certain stretches for travelers going short distances on "ordinary business".

Nor was the president or managers of the turnpike purely private individuals: William Bingham was the corporation's first president, but was also Speaker of the US House of Representatives, and later a Pennsylvania Senator.

The corporation did raise capital for the project by selling stock to private individuals, and its initial sixty-two mile stretch cost $450,000 (an enormous sum for the time). But costs and progress of the turnpike's construction were regularly reported to the Pennsylvania legislature, and further regulation authorized methods of toll collection and specified how legal suits over toll collections would be pursued, and it what time frame.

I mostly point out these details for the Lancaster Turnpike not only for information on it but as an example of other such private turnpikes as existed at the time. There wasn't really a sense of a "public" sector separate from a "private" sector as we understand it today - these companies were something more like quasi-public corporations.

Charles Landis. The First Long Turnpike in the United States

It sounds like this may refer to the Jarbidge Shovel Brigade, which defied the U.S. Forest Service (and numerous federal laws) to open a road on federal land that was allowed to return to nature, making it impassable. This occurred on the Idaho-Nevada border in 2000. I am not prepared to answer your question, but rather, I present a question to you and your friend, asking if this was not the incident in question???

The only thing I can think of that fits your friend’s anecdote is the Iowa River-to-River road, a statewide campaign to grade a road across the state. Of course, the farmers were made much more willing to do the “dragging” (grading of the dirt surface) by a state law directing township officials to pay the farmers 50 cents per mile traveled in doing such work. It appears that signposting the road was done in only a single day in 1910, as a publicity stunt. Secondary and tertiary sources sometimes suggest that all the work on the road was done in a day, but none of the primary local newspaper articles do.

It strikes me that your anti-government friend would be particularly offended by how roads in the 19th and early 20th century actually were built and maintained in some states—by corvée labor. Sometimes called statute labor, this was an ancient system rather like jury duty where able-bodied men were required to give a few days' labor each year working on a road crew. The practice faded rapidly in the US after the Civil War—in part because the quality of work performed was poor. Alabama was the last state to abolish this system, in 1913.

In the early 19th century, there were some limited federal roadbuilding efforts, including the National Road (which got no further west than Indiana) and other important wagon or military roads. The Constitution even explicitly gave Congress the power to build post-roads. But interest sputtered out in the 1830s amid intense national debate over funding infrastructure improvements considered “internal” to the states. In the late 18th and early 19th century, there had been “cross-country” turn-pikes built between Eastern cities by private groups, but few were financially successful, even those offered state land grants, as the railroads would later be. Both national military roads and private turn-pikes soon became unimportant as the railroad came to America and soon dominated intercity land transport. A second wave in the 1840s of private plank and gravel roads, this time feeding ports or railroad centers like Chicago, was not much more successful than the earlier turn-pikes had been.

It’s worth remembering that even in horse-and-wagon days, with railroads unrivaled for intercity travel, each farm or ranch needed some sort of road access. These were typically created and maintained by counties in the US, though some states pushed responsibility down to the township level. West of the original 13 colonies, roads are inextricably bound together with the land ownership pattern promoted by the rectangular survey system, which divided land offered to new settlers into squares, with farms typically 80 or 160 acres in the Midwest and Great Plains.

With all those scattered farmsteads, one very practical response was section line roads, placed on the boundaries between landholdings. For such a prominent part of the geography of the US, there's a frustrating lack of historic resources regarding these and how they came about, but in general, most states west of the Appalachians in the 1860s passed laws enabling counties to declare that a right of way 66 feet wide (33 feet taken from the adjacent land on both sides) along all section lines was a public road. States generally had statutory provisions for compensating landowners for land taken for roads, but this was no problem for land that wasn’t yet privately owned. After Congress in 1866 granted a free right-of-way over unreserved public lands, Dakota Territory made all section lines public roads.

Of course, these rural roads were sometimes rather theoretical: merely the right to traverse open prairie or hard-surface desert, or maybe a wagon track you could make out in early summer. Even in cities, roads generally weren't improved with gravel or asphalt until the late 19th century. The Good Roads movements started around 1890, driven by three factors: the growing popularity of bicycling, the sudden popularity of Rural Free Delivery (mail service to individual farms), and a surprising advocacy by railroads seeking to make their depots more easily reached by farmers several miles distant. In 1892, New Jersey became the first to offer state aid for roads, but the growth in auto use in the early 20th century was the big transforming factor. Most states set up fuel taxes to be used for road improvements, and a portion of that went to counties for improvement of section-line and other local roads. Moreover, nearly all cross-country highways were originally cobbled together from existing roads, often section-line roads, and later improved or rerouted with bypasses of difficult terrain as funding allowed.

Private promotion efforts for cross-country highways used private money, particularly corporate contributions from auto-related businesses, to signpost roads like the Lincoln Highway (or Iowa’s River-to-River route), but private groups only had enough money to fund occasional demonstration projects showing how better roads could be built. The actual burden of improving such a vast network fell to the states, which got increasing federal funding starting in the 1910s and accelerating during Depression public-works programs.

The history of American roads is most entertainingly told in Historic American Roads: From Frontier Trails to Superhighways, a compilation of the illustrated features that had appeared for decades in American Highways magazine, published by the Bureau of Public Roads. I was surprised this afternoon to see how much the Federal Highway Administration's magisterial history America’s Highways 1776-1976 drew on the same material. America's highway experience—including roadbuilding—is the subject of Jakle and Sculle's Motoring: The Highway Experience in America. Highways and Highway Transportation by George R. Chatburn is a look at the subject from antiquity to 1923.

There are quite a number of books about the advent of the Interstate system, but those are for another post. Alas, the only book I have ever found that discusses section-line roads in any detail is Hildegard Binder Johnson's Order Upon the Land, which focuses mostly on Minnesota.