Can I show the timer on an HTML, non-WordPress site?

Yes, you can. The process is this:

Create a WordPress site.

Install Countdown Dynamite to it.

Create a draft post (it doesn't need to be published or visible to others).

Configure the countdown timer as you want it.

Save the settings.

Scroll down and click on the button that says "Get Embed Code to Add To Remote Site >>".

Copy the code that appears.

Paste the code in your remote, non-WordPress site at the place where you want the timer to appear.