For a good understand of the differences I would suggest reading the comments by Michael (of SlimDevices) in this thread:
In summary, it depends upon the devices internal CPU capability. SqueezeBox 3 is less powerful than a Duet which is inturn less powerful than a transporter. The transporter is audiophile grade (and has a price to match) but its seriously impressive.
I suspect the HDX is processing somewhat lower than all three of these players as to my untrained ears I can hear a distint difference. Maybe this is just a software limitation which possibly can be overcome in the future, maybe not, but at the moment its only capable of playing my lowest bitstream flacs at wht I percieve to be a lower quality.
I personally prefer the SqueezeBox3 in their lineup. It looks fantastic, its rock solid and the interface is extremely polished... and yes I do think the Duet sounds slightly better but I regret not buying a second SqueezeBox 3 instead.
On the HDX/popcorn hour, you cannot search by genres, artist, title, year as well as playlist (pre-compiled or made on the fly), random (in each of the prementioned categories or in combination) amongst many other options. And if your hell bent on getting a Duet you also see the album art. On the HDX/popcorn hour I have been only able to play the current title or a complete folder. As far as I am aware you cannot even play in a random order.
I would highly recommending reading a few of the online review or even watch a webcast to get a feel for the product. Its a really good product.