To change color of newsletter button in Squarespace, you can follow these.
#1. First click Page (Main Navigation or Not Linked) where you added Newsletter Button. In my example, it is Page: Canva.
#2. Next, click Site Styles icon on top right of screen
#3. At Colors, click Arrow
#4. On Left side, you will see Theme Color that the section containing the Newsletter is using. In my example, it is LIGHTEST 1
On Right side, click LIGHTEST 1
#5. Scroll down to NEWSLETTER BLOCK, you can change color of newsletter button here.