I don't get as much time with my guitar as I'd like, and I'd like to be "in shape" for the time of day when I can play. Are there exercises I can do while I don't have my guitar that can help improve the strength of my fingers, especially my ring and pinky fingers? Are there exercises I can do to help me move them independently of each other? My ring finger seems especially eager to follow my middle finger.
Something like the Planet Waves Varigrip should help you out.
Get a Powerball to train your fingers and hands. Quite effective.
Many of the answers so far address strength primarily. I'd say the fret hand requires stamina more than raw strength. Many of the exercises people have suggested will increase stamina too, so that's all to the good. I guess I'm just suggesting you be aware of what you're really after.
You also asked about finger independence, and I'd imagine you're interested in general dexterity as well as strength. The Finger Fitness exercises by Greg Irwin are good for finger independence and swiftness. I wouldn't buy his whole DVD series. Just find some of the short videos on YouTube, and you'll get the idea. You can build your own dexterity exercises from the examples.
Since this was brought up from the past, I might as well post my suggestion as well. I use the product line called Grip Master. It's a grip training device which strengthens your fingers considerably, and it comes in different tensions so you can start from whatever current level your fingers can handle.
Stephan's answer is also right in that you want to practice stamina as well. The answer to this is how you use the grip master. It's not just about being able to push it, but how you push it. You will want to push with each finger one at a time so you build up each finger itself, and try to hold it for a certain amount of time. Holding it will bring up your stamina and strength at the same time.
I have thought about this too, from the point of view of an old guitarist who is currently trying to master the piano. My main conclusion is that I am really not convinced that "finger strength" is the crucial thing; what you really need for almost any musical instrument is flexibility and finger independence.
For instance, when you say " My ring finger seems especially eager to follow my middle finger", this is not at all a question of strength, but rather a question of independence, or maybe flexibility.
As a previous answer notes, there are some pretty amazing finger exercises by Greg Irwin available on the Internet that require absolutely no extra equipment, other than the digits that nature has equipped us with. These links to Youtube videos by Greg might prove very useful (or at the very least, entertaining):
There are quite a few more, and I would certainly recommend that any learner of any musical instrument takes a look. You might find them a bit of a revelation.
I've had luck with the Grip Master (and other derivatives) series of hand exercisers. I can easily and quietly use it at my desk at work while mindlessly working on something else.
There's a PDF that Prohands (the originators of these products) that has many exercises available for musicians with demonstrations; but I've found, specifically with guitar the fingertip/small of thumb grip (They called it Finger Tip Pinch) has worked best for me.
It does increase strength, but stamina - as mentioned above - is also a large factor. So doing reps while holding the grip down for 3 to 5 seconds at a time is really where all the benefits come from.
Here's a quick demonstration of my favorite grip.
Another benefit to these is the warm-up time when playing is drastically reduced when used lightly beforehand.
However, with any sort of exercise, too much of a good thing is bad, so trust your hands, do what feels right and think about the motions, grips and movements you make on the guitar; for repetition is key to a musical instrument. Also, go with quality over quantity. Don't be like me and spring for the heaviest tension exerciser they have...