I am looking for a case with a single external bay, front USB 3.0 via motherboard headers, accept GTX560ti length cards, "quiet". Here's the kicker: max depth is 350 mm. Not concerned with max width but anything 250+ mm and I will run into space issues. Height wise under 350mm would be nice but I am ready to compromise.

My own research led me to the Shuttle H67H3 but there are certainly other alternatives.

What I plan to fill the case with:

CPU: i2500k
RAM: 8GB DDR3 1600
GPU: MSI GTX560ti Twin Frozr II OC
Mobo: probably mATX
PSU: something quiet
Storage: With the current shortage of HDD, most likely SSD
OD: BD reader with DVD writer

Thanks for listening!

There are only a few tower cases less than or close to 350mm deep.
Linkworld LC727-10 35*17.5*35cm(L*W*H), unknown quality, no USB 3.0.
Silverstone SG-03 200 mm (W) x 360 mm (H) x 312 mm (D) , limited CPU cooler height, no USB 3.0.
GMC Toast WxDxH 170 x 328 x 420mm , full ATX, unknown availabilty (EOL?), no USB 3.0.

A desktop case turned on it's side could work:
SilverStone Grandia GD06B WxHxD -> HxWxD: 440x150x340mm, 2 USB 3.0 with internal header. 2 3.5" HDD hot swap bays.
Lian Li PC-C50 WxHxD -> HxWxD: 445x167x325mm, 2x USB 3.0 via external header. Aluminum construction.

Cubes:
Antec New Solution NSK1380 WxHxD: 269x201x335mm, no USB 3.0. Comes with a PSU.
Lian Li PC-Q11 WxHxD: 200x326x260mm, tiny, limited CPU cooler height, 2x USB 3.0 with external header, Lian Li has an adapter to convert to internal headers. Officially mini ITX only, mATX boards fit if they are less than 220mm deep, i.e.MSI H67MA-E35, all mATX Z68 boards are 246mm deep. To fit mATX, the HDD tray has to be removed. I use this case with an older S775 mATX board.

All cases except the Antec accept a standard ATX PSU. Take a look at the recommended PSU list, in short Seasonic X-Series, Enermax Modu/Pro 82+/87+, Nexus NX Series, in that order.
Mini ITX cubes like Silverstone SG05 or the bigger SG07 could work, too. Neither one has USB 3.0. Both come with a PSU but neither PSU is really quiet.

I would use the Q-11, Zotac Z68-ITX WiFi mainboard, a Scythe Big Shuriken heatsink and an Enermax MODU87+ 500W or Seasonic X-560 whichever is cheaper. The MSI 560 Twin Frozr won't fit, but a Gainward GTX 560 Phantom will, since the PCB is shorter. The heatsink is pretty good if it's still too loud you could disconnect the fan and strap a 120mm fan on, no warranty voided. You can use a 120x25mm PWM fan on the Big Shuriken to lower the noise level even more. Just remember to mount it in a pushing air from the heatsink so it doesn't work against the PSU's fan.

Thank you for the detailed reply.

HTPC case would work if they could support the weight of the monitor which I doubt.

Sadly, as many have reported here, the Gainward GTX 560 Phantom is not available on this side of the pond. The equivalent (Palit owns Gainward) looking Palit GTX 560 Ti Twin Light Turbo is not available at any Canadian re/e-tailer.

Newegg is OOS for all Palit products and NCIX only has the GTX 560 Ti 822MHZ 2GB for 299.99.

Bottom line, if you want to play, you gotta pay.