Here are some things that I have noticed in trying to improve my squirrel photography. Many of them apply to photography in general. If I think of any other tips, I will post them here when I get a chance and indicate the date that I have added them. If you are an advanced or professional photographer, you may not find much new here, but if you are newer to nature photography, some of these tips might be helpful.
1. The fundamental rule of animal photography: focus the camera on the animal's eyes. This can be difficult with squirrels, being the squirrely creatures that they are, but it is essential that the eyes be in focus. (I have lots of substandard images in which I unintentionally focused on the middle of the body or the tail, usually because the squirrel moved too quickly for me.) Only on rare occasions and for artistic purposes should the eyes be out of focus.
2. You are making an entire photograph, not just an image of a squirrel that is cut out or surrounded by white, so pay attention to everything that will be in the photo. Rather than "taking a picture of a squirrel" you are "making a picture of a squirrel as part of a scene that has lots of interesting and important things," and the parts besides the squirrel (including how the squirrel fits into and relates to them) can make or break the photo. I personally prefer not to think of photos as having "subjects" surrounded by a "background" but rather as entire entities themselves.
3. Pay attention to the light. Being outdoors, you don't control the lighting, but you may be able to move around to catch the light from a different angle, or even just wait a few minutes for the light to change. Also, if this is a place that squirrels regularly hang out, then you can instead try to photograph them at times when the light is good for the type of shot that you want to do.
4. Watch the tails! I can't tell you how many squirrel photos I have in which the squirrel's tail has moved partway out of the photo. For a front or back view of a sitting squirrel, portrait orientation for the camera is often useful to help include the tail. For a side view of a sitting or running squirrel, landscape orientation is usually better. For a front or back view of a running squirrel, an approximately square image may do (if the tail is not sticking too far in any particular direction), in which case either orientation would work. In general, the tail position is important for determining the orientation of the camera. If in doubt, or if the squirrel is scampering about a lot, then you may need to zoom out a bit (or step further back) and then crop later. That's not optimal, but it's certainly better than unintentionally leaving part of the tail out of the photo!
5. Try to anticipate the squirrels' actions. How can you possibly do that? Observe squirrels, a lot. You will begin to notice aspects and patterns of how they move. Do they take a regular path to their tree, or to stash nuts? What does it mean when they flick their tails that way? What (if anything) do they do when they are just about to jump? And so on.
6. (added Jan. 5, 2013) Listen! When you are waiting quietly for squirrels to photograph, you can often hear a squirrel approach (especially from within a tree) well before you see it. You also may hear squirrels communicating with each other too. They have some very distinctive calls!
7. Squirrels have a phenomenal geospatial memory. Those that are scatter hoarders (that bury their nuts all over the place instead of in one big stash) find their buried nuts not by their sense of smell but by their memory of where they put them. So if you see them doing something in one location, watch that location and see if they repeat similar actions in that location on other occasions.
8. If you are photographing squirrels that are on the ground, try crouching or lying still on the ground, in order to get as close to their eye level as possible. This has several advantages. One is that it makes the photo seem more intimate and gives people a view of squirrels that they don't see as often. Another advantage is that it is less intimidating to the squirrels. They have very good motion detection in their peripheral vision (particularly their upwards peripheral vision), and they are much more likely to run away if you are towering above them and/or moving.
9. (added Jan 26, 2013) Move slowly. Squirrels are very good at detecting rapid motion, and it usually (and understandably!) scares them away.
10. If you have the luxury of returning to the same area with the same squirrels multiple times, it helps to allow them to get used to the sound that the camera makes. So click a few times when they are near, not worrying about the actual photo, and soon they will probably become less frightened of the sound.
11. To bring squirrels to an area, try making raw unsalted nuts available to them. Since squirrels are wild animals, I personally prefer not to feed them by hand (although I know that many people do). However, I put food out in the same place (in a squirrel feeder) each morning, and that tends to bring several squirrels to the same place again and again. As mentioned above, they remember locations with food very well.
12. (added Jan 5, 2013) Be a strong critic of your own work. Even many professional photographers will take not just dozens but hundreds or thousands of photographs for each one that they display for others. Learn from every photo that you take, but try to exhibit only your best work. (Of course, that will change over time, but when you select what to display, try to make sure that it lives up to your own standards at that time.)
13. (added Jan 5, 2013) If your camera allows you to change settings in manual mode, use that to your advantage. Decide what type of shots will work well in the light that you have, and set your camera accordingly. One little trick when photographing squirrels is to set the ISO as high as you can tolerate (the highest level where noise does not detract too much or can be dealt with in post-processing). This will allow you a faster shutter speed, which will make your images sharper, particularly if you are hand-holding your camera (which I usually am when photographing squirrels, since they move around so much) or if the squirrels are moving (which is quite often!). One exception to this is if there is plenty of light and your shutter speed is already faster than, say, about 1/1000 of a second. In that case, you might want to use a low ISO to keep noise down.
14. (added Jan 5, 2013) If your camera allows you to shoot a burst of images with one press of a button, use that mode! Squirrels are constantly moving around and doing different things, and a fraction of a second can lead to a very different picture.
15. (added Jan 26, 2013) Be patient, and I mean this in several ways. Be patient in any single outing when you are photographing squirrels; if you miss one shot (or a hundred), don't worry -- there will be more! Also be patient in the sense that you may need to try to photograph squirrels on many different occasions. Sometimes the light isn't working, sometimes the squirrels won't be cooperating, and so on. And in an even larger sense, be patient with yourself. If you try photographing squirrels one day but don't get any photos that you're pleased with, try to figure out how you can improve your photos next time -- you have just had a good practice session. And with practice, your photos WILL improve!
16. Most of all, respect and appreciate the squirrels for the wonderful creatures that they are, and have fun sharing your enthusiasm for them with others.