I am using the Custom Error Response Settings for my AngularJS SPA so that when a user tries to go directly to a page and it gives a 404 not found then it redirects to index.html.
What I would like to know is does this in any way cause a problem for the search engines or would they not even notice the 404 and the redirect to the index.html?
You can (and should) return 200 on those urls.
Based on your tag, I presume you're hosting on CloudFront. You can there set a custom response code 200 for 404 errors, by editing a CloudFront distribution and going to Error Pages tab.

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