(Edited: )

I did some digging for you and found that someone's made a simple white text/black background script for Twitch chat using Greasemonkey. It's an addon for Firefox which allows you to change the display properties of certain elements in web pages. You can also use Google Chrome combined with Tampermonkey.

Firefox may be downloaded here:

Greasemonkey may be installed from here:

Google Chrome may be downloaded here:

Tampermonkey may be installed from here:

The script may be installed from here:

Disclaimer: only install addons and scripts that you trust. The script can be reviewed before installing, just click the "Source" tab. This one has nothing malicious, just changes the text and background colors.

There are a bunch of scripts for Twitch, so there may be one that does it better/have more features, etc. but this will get the job done. I know it's against the rules to cross-post so a mod might be able to move this, but you might belong in the "Feedback" section. Even though this may be a possible solution for you, it is still a good suggestion, and if things can't be changed on the backend, it may be a feature to suggest to Twitch.

Edits: Added Chrome and Tampermonkey links