Do the clients have to set up an account to send the form back to me?

Do the clients have to set up an account to send the form back to me? My
husband filled out a form as a test and said the only weird part was that
he had to click on his google account at the end.

No, your form probably has the privacy settings enabled. 

I just tested this form https://www.jotform.com/form/201088051644046 and I was able to view it without being asked to setup an account.

Set the privacy to public and your form should be good. Let us know if you have any questions or if you need further assistance.