Are there any free image converters that will convert images to 16 bit bitmap file?

Yes, there is one image converter called 'Pixelformer' that can convert a .jpeg image file into 16 bit bitmap file and it is available for free on the internet.
Infact you have three options available while saving any image as bitmap file using this converter. You can save your image as a,

1. 15 bit bitmap file
2. 16 bit bitmap file , or
3. 24 bit bitmap file

All you need to do is to click on the file tab, then go for the import option, open the image file and then export the image as a bitmap image. (You have other option too of saving it in .png etc). Choose the destination for saving the image and click save. Once you click the save button it asks for three choice mentioned above.

The converter can be downloaded for free from the link http://www.qualibyte.com/pixelformer/download.html . It is just a 880 kb setup file that downloads in few seconds. The converter has got similar options as there are in paint. Using it you can additionally create bitmap images using this converter in 6 default sizes or a custom size. Also the images can be created in five different bit (1 bit, 4 bit, 8 bit, 24 bit and 32 bit) bitmap images.