The better question is not “can you…” but “should you.”
And the answer is “no.”
By now, you probably know that you can use an external hard drive with a DIRECTV DVR by connecting it to the eSATA port on the back of the DVR. The external takes the place of the internal drive, meaning that you can’t use both at the same time. So, in order to get your money’s worth, you’d need to replace that internal 500GB or 1TB drive with something larger, right? So, using an SSD suddenly becomes an impossibly expensive proposition if you’re talking about 2 or 4TB of space.
Not only that, the faster drive isn’t going to speed up your DVR, unfortunately. Slow DVRs tend to be a result of slow processors, not slow drives. The rate at which information is read or written to the drive is tied to real time. In other words you can’t write stuff faster than it comes to you over the satellite, right? Yes, you might have noticed the DVR seem to bog down a little bit while recording five things and playing back three things, and yes to a certain extent that is probably due a little bit to the hard drive, but it’s mostly the processor trying to handle eight things at once. A faster drive isn’t going to help as much as you think it will.
Not only that, SSDs aren’t terribly good at the kind of file transfers that DVRs use a lot of… very small file chunks that get read, written, and deleted quickly. That sort of transfer is actually better suited for a good, old-fashioned hard drive.
So, save your money… don’t worry about replacing the drive in your DVR with an SSD.