Did this guy quit, or did I fire him?

I have no opinion about legal issues, firing for cause, or unemployment insurance. I am, however, baffled by the logic some of the posters here are using.

Imagine I meet an employee of mine as s/he gets off the elevator. There are two doors in the hallway. I tell the employee, “The green door leads to the office. You can walk through it right now and go to work. But if you go through the blue door, you will no longer work here.” The employee walks through the blue door.

I cannot see any way in which that adds up to “I fired her.” Nor can I see any relevant difference between that scenario and the OP’s situation.

The employee had a choice: do a specific thing (green door/9:00 A.M.) or lose employment. The employee chose “lose employment.” That’s the definition of quitting.

And if she had a good reason to walk through that other door then, and went back to the hallway and through the green door five minutes later? She’s sitting at her desk, working. If you tell her she quit, you are likely to lose any challenge about unemployment. If she never shows up it is a different story.
What if you don’t tell her which door is which, and she has to randomly choose one? Does she still quit if she chose wrong?

When he said he’d mail back company equipment, he quit.

The truth is, in this regard it doesn’t matter if they were fired or resigned. The outcome for unemployment will be exactly the same. Saying you resigned from a job and trying to collect unemployment makes it much harder to do. If you were fired, and the employer fights it that it was “for cause” then you employee has to fight to prove otherwise. There might be some legal term that defines the separation from the company, but the result will be the same.

It will also be the same for reference checking. If the employee applies for another job, and says he was fired from that job, that doesn’t look good for him. If he says he quit without having a job to go to, that raises questions too. I can see someone moving with their spouse to live in another area and being out of work having resigned from their previous job because the family was moving, because that’s a valid excuse. But it raises a red flag for most employers if you are unemployed, claim you resigned from your last job. They think you were fired and not savvy enough to claim you are at least doing free-lance work in your profession.

The OP’s company when called for employment verification for the employee applying for a job elsewhere, they could give the dates and say the employee is not available for re-hire. I don’t see what the employee did was in any way getting away with something. This is like running your credit score for no good reason. It might very well be the employee has some other issues like substance abuse and was coming up with lame excuses to cover for the real reason of not reporting to work. Or he decided he wanted to be fired to collect unemployment. In any case, he screwed himself over by losing a job, not having another job to go to, and by totally burning the bridges with the previous employer and possible co-workers, and has created a horrible reference for himself.

I’ll second this - give the handle I assume you’re in Oz, the law for NZ is (or used to be) rather similar.

No matter what the agreement between employee / employer breaks could not be “contracted out of” - they HAD to be given, if at all practicable. And it was the responsibility of the employer to ensure that a break was taken, up to and including disciplining the employee for not taking breaks. (which might seem counter intuitive but there you go)

Coming back to the OP again - under the law I’m familiar with, he quit.
BUT - the case will rest on what documentation you have that he tardiness was unacceptable, how this was communicated to employee, and what previous actions were taken.

i.e - you can’t go directly from “no action” to “fired for being late once” (and it is “once” if no prior formal action were taken)

Doesn’t matter. As I said, if she walks through the door, her employment ends. If she makes that choice, she is* ipso facto* quitting.

And as I also said, I’m not concerned with unemployment. I’m only concerned with what makes sense logically. I realize that often has no bearing on what happens legally.

Now you’re just being silly.

Your example doesn’t give enough information to be able to say anything. The link posted above talks about when employers force an employee to quit in one way or another - people in that situation can get unemployment, and my reading is that the law treats them as if they were effectively fired. What is that other door? Is it the door to a rest room? A competitor? A closet? What’s the history here? Why was the choice offfered. (The boss sounds psychotic to me. If she goes through the door is she going to be asked how many lights she sees?)
If not jumping to irrational hoops is quitting, I bet many of us have effectively quit ages ago.
We can generalize the example also. If your desk is messy, you have quit? If you drink three cups of coffee, not two, you quit? If you don’t use the write cover sheet, you quit?

May I ask what kind of business this is?

Software