Where can I find background history on combat patches?

670-1 gives a history, in that it lists every period and country eligible for SSI-FWTS.
The latest versions also removed the minimum time in country requirement, which means you qualify the day you arrive in a qualifying country or area, as long as you also have orders sending you there.
The reg also points out that the SSI-FWTS is an Army award for Army Soldiers, meaning that if you were not in the Army on that deployment, you are not authorized the SSI-FWTS when you join the Army later, i.e. a Marine deployed in 2007 and joins the Army in 2010 is not authorized a Marine SSI for his deployment with the Marines, which I suspect is why your 1SG feels your company needs a class.
Outline your class: History of the SSI and your unit's deployments, highlights of current reg, current list of countries/areas. That should add up to about ten minutes.
Then write up a list of questions that could possibly come up, and write down the answers for those.
Practice the class out loud about ten times, then practice it in front of someone else at least once.

First is to search for information by their formal names: Shoulder Sleeve Insignia-Former Wartime Service. I'll provide you one reference to get you started; check out Para 19-17, Army Regulation 670-1 (25 May 2017)

When presenting information to military, follow the rules of KISS and BLUF.

KISS="Keep It Simple, Stupid" means not to over-complicate the topic by adding more background information than necessary to answer the problem.

BLUF="Bottom Line Up Front" means to provide the most relevant information first, and if you need to expand further, do so after you've provided the "bottom-line up front". It's the professional term for writing a TL;DR "Too Long; Didn't Read." Similar to what I did in this response, I answered your question first (part of it at least), then expanded into further details afterward. So in case you ignore everything afterward, you still have the most important information without having to wade through a sea of text. People can have short attention spans (or limited time), so keep them interested by drawing them in with the answers they wanted, then give them the extra later.

Create bullet points for your main points, meaning, don't write your entire script on the slide and then proceed to read it verbatim. Write a short phrase or word on the slide to remind you of the talking point (and cue the audience of what you are about to talk about), then speak directly from your knowledge of that portion (which requires you to actually know the topic through personal research).

Look at AR 600-8-22 Awards, AR 670-1 and DA PAM 670-1 Uniform Wear and Appearance, Those will give you a good start. In those regulations will be lists of supporting regs and messages that define periods of service and other info on the combat patches.