Most likely one of the two things:

First check: is the page published? Open the page as editor and look for the Published / Unpublished toggle. Flip it to Published and test in a private/incognito window where you're logged out.

If it's published and still 404s check public access to the underlying model. This applies if the button leads to a custom form/page rather than a standard website form snippet. 

If that's the case, you grant the public group access to the model(s) the form uses from Settings → Technical → Security → Access Rights (with developer mode on), add a rule for the Public group on the relevant model.

Let me know if either of those work.