Why did Native Americans never unify into a single country?

Because they never advanced beyond hunter-gatherer society

In order to have a country (and latter nation) you first need to have permanent settlements. This is important because monarch, council, senate or whatever ruling body you have, could only execute their power over finite territory. Historically, there were examples of vast empires (Romans come to mind) where orders from center traveled for days. Still they had starting point and ending point for those orders. But communication between two tribes that keep moving is almost impossible if they move more then few days of travel from each other.

Now, to have permanent settlements, certain level of agricultural technology is a must. People need to have enough to eat, and to do that they need to cultivate crops and raise livestock. Mesoamerican Indians did have that and therefore they did manage to organize countries with central power like Aztec Empire or Mayan city-states. Note that even in those cases agricultural technology was not developed enough to avoid collapse , possibly associated with overuse of soil do to overpopulation around great Mayan cities.

However, North American Indians never achieved agricultural level of their southern brethren. At best, North American tribes were semi-sedentary and supplemented their diets with maize and occasionally beans. However, yields were limited, plots of land were small and whole agricultural business bordered with gathering which was traditionally female occupation. Having better yields require labor intensive work like plowing fields (usually with domesticated animals unavailable to Native Americans) which is typically male job. This transition from female (small scale) to male (main source of food) agriculture never happened in North America until the arrival of Europeans.

Consequently, since North American Indians depended on hunt as main source of food, they required fairly large hunting grounds for each tribe and moved a lot. Other tribes were viewed as potential competitors, and clashes were not uncommon. As a rule, even when they were not in war, different tribes tended to keep themselves on a distance from other tribes which precluded unification. Therefore, except occasional confederation of tribes, they didn't advance much on organizational level .

I recommend you read Guns, Germs and Steel form Jared Diamond. It's not 100% accurate, and there is some controversy over it, but it gives you a good idea why some people got a better hand in life than others.

Why, in over 10,000 years of occupying North America did they never advance from a tribal culture and form a single united country

Some tribes did form cities. But they didn't progress beyond that level. Because they lacked what is necessary for larger societies: no wheel, no large mammals and no writing.

This also applies to some extend to the Aztec and Inca empires - which were pretty big. They proved you don't need all of that to form a large empire. You can do that without large animals (llamas can't carry that much and cannot pull carts) and without writing. But only without competition. Once the Spaniards appeared they couldn't withstand them. It's a bit like DOS meeting Win 10.

Every society or group of societies evolves differently. In the Americas for example the wheel was never used seriously. Some artifacts or children's toys have been found with wheels, so they knew wheels existed. But they never used it. Which makes transportation of goods very difficult. Add to this no big mammals. No horses, no cattle. Which means you can only carry something on your back. That limits anything you want to do. Commercially and militarily. That fact alone can explain a lot, I think.

But they lacked much more. Writing for example. No wheel, no large animals, no writing, all this adds up.

Another problem was no real seagoing ships. In South and Central America most or at least many rivers are difficult to use for transportation (not navigable, or going the wrong direction). No Indian civilization had seaworthy vessels to compensate for that.

There's a bit of a false premise here, in that there were Native American empires. I suspect you meant why weren't there any north of Mexico, as there is no way you could overlook the Inca, Aztec, Maya, Olmec, Zapotec, etc etc, and my answer is built around that premise.

Life in America is Hard

When the Europeans first came to America, they had trouble surviving, even with Native help. The modern life hides this, but most of America and Canada is very difficult to live in without technology. The winters are brutal. Natural resources are minimal. Animals to domesticate are far and few between. It's far easier to advance science and build empires when your nation exists in a state of surplus, and that just doesn't happen north of the Rio Grande (ok, there are a few exceptions, but not many) without technology. It's a catch 22; they needed technology to improve their economy, but they needed a better economy to develop technology.

Where Would You Spread, Anyway?

America has a funny geographical feature: it has massive mountain ranges that run exclusively north-to-south. The Coastal. The Cascades. The Rockies. The Appalachian. North America is blessed with some pretty major mountains, being a geologically newer continent and all, but north/south migration is far harder than east/west migration. When you start moving north/south, your climate changes quicker. The crops you are used to won't thrive. The animals you take die. The weather patterns are unfamiliar. The seasons aren't as long as you're used to.

Most of the US and Canada has a naturally harsh climate and significant geographic barriers to east/west expansion. It is no surprise that they struggled to grow into significant empires.