Why did the US military never adopt the Henry repeating rifles?

I can't speak for the Henry specifically, but repeaters in general faced a lot of resistance by nearly every major power almost up until the advent of smokeless gunpowder in 1886. This reluctance to adopt repeating weapons for standard infantry rifles makes a lot more sense once you start looking into it - the firepower of the individual man may have been improved, but it became a logistical nightmare in an era where logistics were a major issue for armies. Remember, at the time, the standard infantryman was only carrying 40 rounds on his person, and a "heavy" load given to vanguards for attacks was a whopping 60 rounds. Even in a period where the rudimentary nature of ammunition meant that the components (percussion caps, black powder, cast lead bullets, and paper for cartridges) was fairly easy to make, running out of ammunition was consistently a major concern for armies of the era, and increasing potential fire rate only made ammunition concerns worse. Magazine rifles like the Henry only added to concerns, as it was feared that soldiers would fire off un-aimed shots in a panic and waste ammunition. This is a concern that actually carried on into the 20th Century, particularly in the US Army and British Army, in the form of the magazine cutoff.

For the US and Britain, this concern over logistics was particularly important because of the nature of the campaigns they expected to be fighting. With large "uncivilized" frontier regions being the regions where both nations had the majority of their action happening at the time, logistics became supremely important, as railheads and supply outposts may be very far away and the only ammunition available was going to be whatever a unit could carry along on its own. The Civil War offers a significantly different dynamic than the frontier warfare the US Army was used to fighting, but even as the war took on plenty of very modern aspects, logistics remained a critical chokepoint for armies. Chickamauga stands out in particular, as one unit was armed with Colt Revolving Rifles (21st Ohio). Here, the 21st Ohio performed incredibly well and managed to hold their ground even as the army was collapsing around them, but they began to run out of ammunition during the battle and only by having other units bring in ammunition were they able to continue fighting. One important thing to note here is the kind of ammunition the Colt rifles used - unlike more modern repeaters, the Colt Revolving Rifle still used paper cartridges and was at its core an enlarged cap-and-ball revolver. So while it could benefit from the simplicity of the ammunition like the muzzle-loading rifles of other units, its fire rate was still high enough to cause logistical problems for the 21st Ohio.

And that brings us to the Henry in particular. The Henry added to the problems of magazine rifles with its cartridge. The Henry's .44 rimfire cartridge was very weak even by the standards of the time, and by modern standards was alarmingly underpowered. Modern standards for "effective range" based on the energy of the bullet put the round's effective range well under 100 yards at a time when infantrymen were regularly expected to be engaging targets out to 500 yards (and beyond, if we start looking at post-Civil War weapons). Compared to the Spencer repeater it was a contemporary to, the Henry had half the muzzle energy. When we look into the manufacturing of ammunition, it only gets worse. Rimfire ammunition is finnicky and somewhat unreliable, as it requires an even distribution of priming compound around the rim of the case. Even today, that's a major source of reliability issues among rimfire ammunition, and it'd be even moreso at a time when this technology is just coming into service. The case itself is another hurdle, as it requires facilities capable of working with brass. I'm unfortunately not familiar with the specifics of .44 rimfire of the time, but the specific method of manufacture would have a major impact on ammunition supply. A lot of early brass cartridges were rolled brass (just rolling a flat piece of brass into the cylindrical shape vaguely similar to paper cartridges), which is a more difficult and time-consuming process than the drawn-brass method used today. Drawn brass is the superior method and as far as I can tell is likely the method used by Henry, but supplying the entire US Army with this ammunition was going to require the procurement of a huge amount of new equipment for working with brass.

And now that gets us to the pressing matter at hand - the Civil War. The Civil War was an immense strain on resources, and US arsenals were working as hard as possible to churn out rifles and ammunition. Shortages were such that even the Union, with its colossal industrial capacity, had problems standardizing infantry arms around a single model. A number of private firms did market cutting-edge weapons to the US Army, and many - including Henry - succeeded in securing orders. However, none of these firms were in a position to produce the kind of volume necessary to re-equip the US Army as a whole, and their expense compared to traditional muzzle-loaders would have made the prospect of such untenable. Because of that, repeaters generally ended up being non-standard weapons that saw more use with cavalry, where effective ranges of weapons and logistical concerns were less pressing issues, and generally only equipped whole units if the unit itself (or its commanders) were willing to pay for the weapons and their ammunition out of pocket. The Henry did see some service with at least one infantry force (one unit at Cold Harbor had them, if I'm remembering correctly), and it did make an impression when it showed up. But when the war ended, the Army fell back on the concerns I described above and instead moved towards a more conventional single-shot breechloader - the Trapdoor Springfield.