The situation described with flying squirrels is known as parallel evolution. It occurs when two related species split from each other, evolve in different places and circumstances, yet end up developing many of the same traits. When two different species share a lot of traits, it's known as morphological similarity. When two completely unrelated species develop morphological similarity, it's known as convergent evolution. It's sometimes impossible to decide which type it is because we don't have complete knowledge of the evolutionary record. We have no way of knowing how closely two species were related millions of years ago.
The simple reason for parallel evolution occuring is that similar environments and similar population pressures do indeed lead different species to evolve similar traits. A successful trait in one place is going to be successful in another. But that doesn't really tell the whole story. After all, there are millions of species on Earth, and a lot of them don't look anything like each other. Why do only some species exhibit parallel or convergent evolution?
It has to do with the way natural selection works. A species can change from one generation to another because of mutations to its genetic code or recombination of genetic information by sexual reproduction. These genetic changes show up as new or altered traits. A mutation might cause a species of bear to have much lighter coloring on its fur, for example. Traits that give the organism a greater chance to survive long enough to reproduce are more likely to be passed on to future generations, while less successful traits won't be passed on as frequently. Thus, over time, the average of the traits across a population of organisms shifts -- the most beneficial traits show up with much greater frequency.
Eventually, these accumulated beneficial traits make an organism very well-suited to function within a certain environment. This is the species' ecological niche. The animals have adapted to live successfully within that niche but would probably do poorly outside it. A polar bear's niche is at the top of the food chain in the cold, snowy climate of the Arctic. A polar bear that tried to live as a grazer in the African savanna wouldn't fare well.
The organisms most likely to exhibit parallel or convergent evolution are those that occupy similar ecological niches. The savanna of Africa and the plains of North America are similar environments -- slightly arid, flat and covered with grasses. The same niche exists in both places: large, herbivorous mammals that live in herds and graze on the grass. Wildebeests and North American cattle evolved far from each other, but they have incredible morphological similarity. Neither species evolved into polar bears -- that wouldn't make sense. Natural selection reinforced the traits that made those species successful within their niche. Since the niche was the same, it's really not a big surprise that the species look the same.
Some convergent evolution doesn't depend on ecological niches because the traits are very advantageous to a wide range of organisms. All carnivores, regardless of where they live, have evolved sharp teeth. Birds, bats and many insects have evolved the ability to fly. They all fly in different ways and for different reasons, but flight is so beneficial it shows up all over the place.
Parallel evolution is fairly common at the morphological level, but what role does the underlying genetic process play? Let's find out.