would Gregor's boss really fire him for being late?

It sounds like he would. Gregor is "sick" in bed. When his boss, the Chief Clerk, hints at discipline, Gegor seems to spring into action.

"Your turnover has been very unsatisfactory of late; I grant you that it's not the time of year to do especially good business, we recognise that; but there simply is no time of year to do no business at all, Mr. Samsa, we cannot allow there to be."

But Sir", called Gregor, beside himself and forgetting all else in the excitement, "I'll open up immediately, just a moment. I'm slightly unwell, an attack of dizziness, I haven't been able to get up. I'm still in bed now. I'm quite fresh again now, though. I'm just getting out of bed...