The short answer is that you would add CSS to the page where the form is embedded (i.e. your client’s Squarespace page).
The long answer is that you may run into problems with CORS and also with the form’s class names and HTML structure changing over time, which will break your CSS.
I’d contact Airtable support directly about your question to get guidance about these two issues.
I had a similar issue, mainly cause I hated how it looked. You can use a Stacker or miniextenstions to create a better UI to imbed on the Squarespace site.
The easiest solution (at least for us) was to just use the open new window function and have the button just share the form via the share url.
If I find out a way to use the CSS ill let you know.