The circuit is AC-coupled, so it doesn't technically require a dual-voltage power supply. If you replace the -12 V references with the ground side of a single-ended power supply, you'd just need to work out some way to get a virtual ground in order to bias the input signal up between the I/O limitations of the buffer/amplifier circuit.
The reason people use dual-ended power supplies with op-amp circuits like this is specifically in order to avoid the hassle and problems of virtual ground circuits. There is no difficulty-free solution, only a choice of which bag of problems you want to accept.
Can I use a 24V DC power supply and then use a voltage divider (maybe with a zenner diode) to create a virtual ground?
That's one method, but as you'll see by reading the article linked above and the articles linked from it at the end, it would be facile to describe this as an easy solution.
Your idea is closest to that of the CMoy virtual ground criticized at the top of the article, only without mention of the rail caps, which mitigate a lot of its problems, but not all. You can do a lot better, which is the subject of the rest of that linked article.
Can I use something like an LM7812 to step the voltage down from 24V to 12V and use that as a virtual ground?
Yes, but TANSTAAFL:
It is only an exact ½ rail virtual ground to the extent that the 24 V supply and the LM7812 are accurate. If the input supply is 5% high and the half-rail regulator output is 5% low, the virtual ground point will be off by roughly 10% of ideal. Depending on conditions, the virtual ground circuits in the linked article can be more accurate than that.
It burns half the rail voltage to achieve this. It'll throw off a fair bit of heat. Probably not enough to require a heat sink, and certainly not enough to require forced-air cooling, but...well, it isn't very engineer-y. Sloppy. Wasteful.
Or is there a more preferred way to accomplish this that someone can point me to?
Lots. Even if you collect all the ideas in my article and those I've linked to, it is still only a subset of the ways people have invented to tackle this very problem.
The most important thing to keep in mind when pursuing all of this is that voltages are relative, not absolute. There is no "12 V", only "12 V with respect to X", where X could be any other voltage potential. Any. Most likely it's 12 V above some ground point, but it could just as well be 12 V above the output of a 500000 V electrical generation plant. Even if you discard wild possibilities like that, you've got many mundane possibilities, like which ground; all "grounds" are not equal.
Also keep in mind that the only reason this circuit you're referring to recommends ±12 V is to make enough room between this op-amp's rails for the input signal to get through and be amplified. If the input signal could be as high as 0.1 V peak-to-peak and the output is therefore up to 1 Vp-p, you could probably push this circuit down as far as 12 V single-ended, depending on the load characteristics. That is an informed guess based on testing.
Keep the load in mind: as those same tests show, if this is supposed to drive 50 Ω input circuitry, you may indeed need 24 V single-ended or ±12 V.