Your old system may have had some power issues. For example, if the computer had only so much RAM and so fast a CPU, it could be running close to its limits while you were defragging. That would easily cause a crash.
Not only that, but your old system could have been clogged up with fragmented files, so defragmentation had become a task requiring a lot of energy.
Your new computer is hopefully faster and more powerful. That is one reason why you'd buy a new computer, isn't it?
So it should be able to handle defragging. You could even consider it a bit of a test. (I'd be astonished if anything went wrong, however.)
?You should realize that a crash is not a bad thing in itself. At worst it should be considered a nuisance. A blue screen actually provides helpful information. But even a black screen crash lets you know that you probably need to modify something. It definitely alerts you to the general maintenance requirements, such as CCleaner and, if you use it, TuneUp Utilities. Do you have the freeware SpywareBlaster? You should. You have a good anti-virus? You can use the freeware AVG or any of half a dozen others. And, of course, is your system defragmented?
Apart from that, there are more settings you can learn in the BIOS. Take a look at your computer manual. If you don't have a copy, look up your make and model and download a .PDF of your manual; then take a look through the section about the BIOS. You don't have to do much there, but occasionally there are tweaks that make your computer operate with greater stability.
And, always, run a strong RAM. If your new computer came with 1 GB of RAM, can it take 2 GB? That might cost you the better part of $100, but it is a good investment. RAM "thinks on its feet" and is exactly what your computer needs to be able to handle a maximum work load all at the same time. It is analogous to your own "working memory" or "short-term memory". Your old computer was very likely hungry for RAM.
Especially with crashes, get away from the common fear of these things. Instead, read computer forums and learn to understand what is happening. A car can let you down on the highway, but your computer is not going to do that. It's not going to blow up, and it won't suddenly lose all your files. (Er?especially if they are backed up!) So don't panic; don't be afraid of the machine. Your whole computer is dedicated to working FOR you, so adjust your own perspective to accommodate it with greater understanding. I think you are already off to a good start because you've discovered these Piriform products, which are very good; and you might also look at other freeware on FileHippo, which can be trusted. In the upper right corner of FileHippo, you can adjust the setting to "only freeware", by the way. (No, I don't represent either of them.) Good luck.