why do you suggest using a screen with your bong?

The thing that makes the water dirtiest is the butane from the lighter you are using.
If anything I'd suggest not using a lighter.
But I change my bong water about every 10 sinkers anyways.
Also if you are truly concerned about your water, i'd suggest investing in an ashcatcher (also known as a perculator, but first intended to catch ash, just for the record)
I got one for about 50 bucks, looks nice, filters the smoke and extra time and also keeps your main water from getting ashes in it and also less butane goes into your main water and mostly all collects in the catcher so you have to change your main water less often and can change the catcher which is a lot faster.

As far as your question, I guess you could use tin foil, or look around in the kitchen for a flour sifter and cut a piece out.
I dont know if you have that sort of thing though.
Other than that I cant think of anything that would be a much healthier substitute