There is a lot of anecdotal evidence but no hard core data to support the claims. It would certainly make sense that creatures like snakes and dolphins should have an ability to sense smaller seismic events preceeding an earthquake. After the quake in Italy some studies were done on toads based on the fact that their breeding stopped shortly before the quake, they fled, and then they came back. But as sceptics would say, could be due to other factors. I believe it though because animals have many more senses than we do and if we can sense a storm or a weather change coming, somebody should be able to sense an earthquake.
I watched something on discovery channel about homing pidgeons. Apparently they have a buildup of iron in a special part of their inner ear, in a way that it detects magnetic north the same way our inner ear detects gravity. So they have 7 senses, and this may be common in birds, especially migratory ones.
So if a bird is proven to have 7 senses compared to our 6, then it's quite possible that some animals can sense other things like fields, or subsonic waves etc.
There is also a very strong human tendency to believe that things that happen in association have a cause-and-effect relationship, even when no such link may exist, or where the link is an indirect one.
For instance, animals might respond to weather changes in general, which sometimes may herald the coming of disastrous events like tornadoes. People may therefore conclude that these animals can predict tornadoes: that may even be true, but it should only be believed if a credible body of evidence can be collected to support it.
"Animals" is a really broad designation. Elephants, for instance, can detect subsonic signals and they have big feet, so they might be more inclined to detect the primary wave of an earthquake. But then again, a primary wave traveling through the air will not reach the elephant as fast as the secondary wave traveling through Earth.
Note that the "instincts" of the animal aren't much different from ours. Once we detect a catastrophic event we run, freeze, etc, whatever our adrenal response is.
The significant part is the detection, not the instinct. Do animals have the physical apparatus necessary to detect the signal? That should be the question.
In case of earthquakes there seems to be a indirect relation to animals behaving differently shortly(relative, could be days.) before a, or better, some earthquake cases.
One indirect link seems to be that rocks that are under high compression release electricity (think it was the static kind) when there fractioning under the pressure. And it is this side effect that some animals react on.
Think there are more of these kind of cases where a preceding, yet not know or fully understood by us, effect triggers a flight reaction in animals.
Now considering animals in general have higher quality of (normal) senses. And they are not burdened by a brain that thinks to much. They just react on things that make them feel uncomfortably.
There might be a evolutionary part playing a role here to. (in relation to the flight reaction of animals on certain natural effects.)
The argument that they 'directly' 'know' or 'feel' that some natural calamities is coming or is in progress is a bit of a stretch. (to put it mildly)