I play the game in 1360 x 768 as I have a widescreen laptop, but there are black borders down the left and right edges of the screen. There are no borders along the bottom or top of the screen. But how can I get the screen the stretch left and right to fill the whole screen?

Modify your dosbox configuration file or your graphics card properties if you want to make the aspect wrong.

The game runs in 320x\640x resolution so obviously your widescreen monitor cannot display properly so that's why you have black bars.