Their (bad) legal argument was. He refused to follow work instructions, hence he did not perform his work as required, hence we can fire him "op staande voet" which is an on the spot firing that does not require a long process of attempting to improve employee performance. They needed a 'op staande voet' firing because the long process takes months. The justifications for this are limited to things like "theft or fraud", "Threatening or causing Grievous insult", "repeated innebriation" (called out specifically in law), or "refusing a task".

That final cause "Refusing a task" ,or in dutch, quoting from the law: "wanneer de werknemer hardnekkig weigert te voldoen aan redelijke bevelen of opdrachten, hem door of namens de werkgever verstrekt", can be translated to insubordination. A full translation of the dutch phrasing would be: "when the employee persistently refuses to comply with reasonable orders or instructions issued to him by or on behalf of the employer". Hence the company argued that "turn on your webcam all working day" was a "reasonable order or instruction issued by the employer".

Note that the story mentions this was during a "Corrective Action Program". That sounds like they were already in the firing process. Which starts with a process of notifying an employee their performance is not good enough, and requires they are offered a chance to improve. This 'offering a chance to improve' is usually done by putting employees on a 'performance improvement plan'. I believe many US states have similar procedures.