Self-employed people do not have the employment rights and responsibilities that employees have, as they are their own basically their own employer.
Being self-employed, therefore, means forfeiting the right to: ***** ***** pay, maternity, adoption and paternity leave and pay the right not to be unfairly dismissed, statutory redundancy pay the National Minimum Wage and so on.
If they were an employee, and they made those comments and just walked of site, it could be questioned whether they have resigned or not. If they had less then two years service then no problem, as unfair dismissal wold not apply. However, 2 years or more, it is safer to obtain their resignation in writing.
Now, back to them being self employed. As stated above, as self employed they forego a number of workers rights. Therefore unfair dismissal would not apply.
There walking off the job can be treated as gross misconduct just as it cold for an employee.
But as self employed, you do still have the option to see if they have quit, if they say no, then your choice if you want them back having waled off site.