My question is if you shoot JPEG in color and then use post processing to convert to black and white, will you get better black and white images that way?
Keith, it depends on how skilled you are at making B&W images from digital captures.
"I wouldn't trust the camera to create the B&W" is a commonly-held and dogmatic (
in my opinion 
) opinion.
"Channel mixing is superior to greyscale conversion" is also a commonly-held and dogmatic opinion.
What you can make with channel-mixing you can usually do also using curves tool (unless also specify 'preserve luminosity') which then will allow some extra/unreal effects.
You can shoot B&W JPEG using physical filters on your lens and you can also even apply digital color filtering even to monochrome images in certain apps that will change the luminance values even without actual color being displayed. You will lose some light using physical filters but often that isn't a big problem unless you are working very low light. (In fact turn that to your advantage and use a filter to suppress light in some situations; allowing wide aperture in strong light etc).
There is a Cambridge in Color article online somewhere which concludes that greyscale conversion is actually the most visually accurate representation of an otherwise color scene.
Some cameras make great B&W straight out the camera. Others not so much. IMO, if you are good with using curves, a straight desaturated image works fine. Especially if you also use an editor where you can duplicate the image and render the second layer in Overlay mode (~50-80% opacity should work). This gives the image a very dense, 'glossy' look.
Some people desire a deep black point in there monochromes (and feel everyone else should have to). Other people only render true blacks where the scene in reality possesses that. IMO people that grew up shooting film appreciate tonal gradation in shadows yet it seems many digital photographers want to render images higher contrast. Just an impression I get.
Using levels tool, lifting the output slider for the black point will soften shadows (and can mask slight posterization ...incidentally, camera picture style setting for JPEGs: some cameras seem to posterize shadow if the picture style contrast setting is too high. Even with in-camera JPEG, set the picture styles low contrast and low sharpness and do those in post).