Compatibility for WPEngine and environments without exec()

The plugins uses this PHP library to handle the compression: https://github.com/mrclay/minify

You’re correct to point out that this uses exec to run the java based compressor. This is not ideal – i’ll have a look at doing this a different way.

It appears that the Java compiler isn’t always used. Would it be possible to configure options to ensure that the it’s never run?

It looks like those files aren’t used by your plugin at all. Default minifiers are specified here:
https://github.com/mattheu/MinQueue/blob/458b4173d5871dd298bf48420a295c4e4f799ad3/PHP-Minify-Lib/Minify/Controller/Base.php#L74

Without another way to specify options, the defaults can’t really change. You could either remove those unrelated files or provide a filter to send options like so here:
https://github.com/mattheu/MinQueue/blob/458b4173d5871dd298bf48420a295c4e4f799ad3/class.minqueue.php#L433

$data = Minify::combine( $srcs, apply_filters( 'minqueue_minify_options', array() ) );