I too would recommend ImageMagick. It has a command line option to remove EXIF metadata
mogrify -strip image.jpg
There are plenty of other tools out there that do the same thing.
As far as recompressing JPEGs go, don't. JPEGs are lossy to start with, so any form of recompression is only going to hurt image quality. However, if you have losslessly encoded images, some encoders do a better job than others. I have noticed that JPEGs done with Photoshop consistently look better than when encoded with ImageMagick (despite the same file size) due to complicated reasons. Furthermore (and this is relevant to you), I know that at least Photoshop can save JPEGs as optimized which means they drop compatibility with some stuff that you probably don't care about to save a couple of KB. Also, make sure you don't have any colour profiles embedded and you may be able to save another couple of KB.
