Use metal boxes.

bus_driver...

Is the point that I can certainly put screws through a metal box?

Jim

The plastic box is not reinforced to be attached through the bottom, making it susceptable to being damaged in the process or in routine use. This wouldn't be a problem with a metal box.

It might be possible to nail in an extra stud where you wish the box to be and nail the molded box on the side of that stud.

Hey there bus driver.

I have the sheetrock off of the wall from 66 inches down to the floor. So it is "new work" in the sense that I can see the studs from 66' down. But I want to avoid having to do the extra framing (even if I could) because I have about 12 of these to do.

Then (as bus_driver and randy both said) use metal boxes.

"1) Could I run a 2x4 horizontally between the studs and attach the box through the bottom of the box so that it is still vertical?" Yes, fasten the box through the back, bottom or side as you prefer. Watch for not blocking the cable entry that needs to be used. Watch that the combination of 2 x 4 plus the box does not result in the box
protruding too far from the wall. Use two 2 x 4, one above the box and one below, and old work boxes if necessary. See the link. Use the one-screw ear for the two 2 x 4 plan Page A6 at the link. It offers more adjustment. Boxes at Page A8 and following. Use #4 x 5/8 flat head tapping screws
in the ear holes to hold it to the 2 x 4s.
"3) Would it be possible to use a very flat head Phillips screw?" I prefer pan head inside the box as they offer less change of conductor insulation damage from sharp edges.
If all this is not yet understandable, hire a pro.

Here is a link that might be useful: Boxes

bus_driver

Ok, great. Thanks.

I would fasten from the bottom. Will leave room for entry for the wires. I need the box to protrude 1.75 inches.

Here is the screw I would use. As an aside, these are the handiest screws ever produced.

http://www.strongtie.com/products/connectors/SD-wafer.asp

NM boxes must have the fastening means EXTERNAL to the box unless the box is a listed assembly with internal mounting hardware. they do make listed boxes for exactly what you are trying to do, arlington makes them.

aifittings.com/catalog/fan-specialty-boxes/one-box-non-metallic-outlet-boxes/f101h

but you can't just put screws or nails inside a normal NM box. use the box above, metal boxes, or what bus suggested about 2 horizontal studs and an old work box.

Thanks, everyone for the input. I got horizontal boxes from Allied Moulded, which I will mount vertically, attaching the two screws from the bottom into a horizontal cleat that I have nailed between the studs. So the height is fixed, but I can position them anywhere I want on the cleat. Works like a charm.