I have had the same problem. (And a google search suggests many others as well.) I have asked SquareSpace on several occasions as this is a really inconvenient problem for small business owners who get inquiries from potential customers through their website, which seems like one of SquareSpace's main market segments.
The workaround: Create a new form. Don't delete the "subject" line. When a user makes a form submission, the text that they enter in the subject line will be included in the subject line of the email you get to your gmail account. So unless you get multiple form submissions with identical text in the 'subject' box, gmail will recognize them as separate email threads and won't merge them.
For example, if you didn't have a 'subject' line in the form and three people submitted inquiries on the same day about your dog grooming business, you would get ONE email thread with three separate emails from three separate people. If you have a subject line, you'll probably get three separate emails in three separate threads, one saying "New Inquiry - Groom my dog", another "New Inquiry - Dog grooming request", and a third "New Inquiry - Are you available?". Of course, if you happen to get two people who both send inquiries on the same day and enter the exact same text in the 'subject line, you'll still have the problem with their messages getting grouped into one thread. But this workaround helps reduce how often it happens. It also explains why every SquareSpace user isn't up in arms about this and how annoying it is...only ones who created forms and removed the subject line field.
If anyone finds a way to use this workaround to permanently solve the problem, do let me know! If there was a way to automatically include a date and time stamp in the subject line, for example.