If a hard drive shows the first sign of a problem, I back it up and plan on retiring that drive soon. I've retired I think 4 or 5 drives over the last 5 years or so...about one a year. I consider one drive a year to be the cost of ensuring I'll never lose my music.
What are the warning signs of a dying external HDD? Are there diagnostic tests that can be run on one? I used to use the old chkdsk utility back in the DOS days. Is there a similar animal now? What about things like disk defrag or disk cleanup?
To be honest, I'm not sure if there's any software that helps diagnose a drive failure. I think the programs you mention help to diagnose bad sectors on a HDD and move data away from those sectors. The issues I'm more concerned about involve the mechanical failures of the hard drive that made a complete drive un-readable. These are catastrophic failures and can't be recovered from without sending your drive to an expert and having them somehow perform a miracle to get your info back (at a high expense).
Keep in mind though that not all drives give you warning signs before they fail. A drive motor can stop without warning. But the most typical warning sign I know of is a drive that starts to make a clicking noise. I've had that happen to me on two impending drive failures. The first time I didn't know what the issue was...even though I should have realized that it was time to back up my drive and that it was about ready to fail. The drive continued to operate for quite a while with the noise, but eventually failed.
A second warning sign is that the drive would not mount consistently every time I connected it. This actually was a second symptom of the above drive while it was clicking.
I third warning sign is probably related to the first, but it's to feel your drive and feel if the drive has any odd feel...if the drive feels like it isn't spinning smoothely or if they're some odd bumping or grinding feeling to it. Sometimes, you gotta know what your drive feels like normally though in order to be able to 'go by feel'.
I'd say none of these are definitive methods, but they can all be helpful in anticipating if your drive is on it's way out.
After learning my lesson the hard way (I lost a bunch of digital pictures that I didn't realize I hadn't backed up) for the first drive failure, I've tried to be as proactive as possible with drive management. So, when I hear any extra clicking, for example, that's when I realize it's time to run out and buy a replacement drive, if the data on the clicking drive isn't backed up.
If the data IS backed up, I might wait until that drive fails, rather than just retiring the drive. It probably just depends on the info I have (weither to wait or buy a new one at first warning sign). However, if the drive does finally fail for good, I'll make sure and buy a replacmenet right away because once I have a failure, that means that I have data that's no longer redundant. Obviously, once I get a replacement, the first thing I do is re-back-up the data from the existing good drive to restore redundancy.
I'm not a huge fan of buying the most expensive drives out there. Personally, I've bought expensive drives and cheap drives. They all fail and I don't see any evidence that the expensive drives last any longer than the cheap drives. Of the 6 drives I've got downstairs, a couple were el cheapo and they're still spinning five years later. So I tend to just go with whatever is available for a reasonable price and just stay on top of it. Fortunately, I've never had a drive fail soon after purchase, but even if I did I would still use the same strategy as above. Like I said, one new drive a year is about what I plan for as an average.