To change the Header button color in Squarespace, you can follow these guides:
#1. First, access your edit mode
Suppose your site is: elf.squarespace.com
You can add /config after to access edit mode, so the URL will be:
elf.squarespace.com/config
#2. Next, click Edit on the top left of the site.
#3. Click Edit Site Header
#4. Click on the Button and then click on the Edit icon
#5. On Design, You will see which style the button is using: Primary, Secondary, or Tertiary.
In my example, the button is using Primary style
#6. Exit edit on the left and choose the Site Styles icon on the right of the Header
#7. At COLOR THEMES ON THIS PAGE, you choose one of the color themes
#8. Scroll down a bit at the Button, change the Primary Button Background and Primary Button Text color
#9. Finally, click Save on the left of the screen to change the Header button color
