Track every minute or boss will mix facts up: tells you that you've been 2 hours late and stayed only 2 minutes over time

Yeah my old job, DM was at our store during week leading up to inventory. He and SM said we were all going to stay after closing one night until the store was fully recovered. 
The other sales associate besides me refused saying she had things to do. SM is saying he can't believe her, she prob didn't even have plans just lazy, etc after she had left. I told him, "well you can't expect people to stay late without notice.... I stayed, but normally I would expect to be notified in advance. Even if I didn't have plans I still don't like my hours being changed last minute like that."
He says that technically it's company policy that closers are supposed to have the store fully recovered each night (true) and we're supposed to stay as late as necessary to get it done (huh?).
First of all, you get in trouble for even clocking out 15 min late because they're so strict on payroll so I didn't see how that could be true. Second, that seems like it might be illegal to force someone to stay on the job past scheduled time?

And yes, I had been written up for being just a few miutes late before.

/r/antiwork Thread Link - i.redd.it