RESINATION
you can't refuse a resignation. Resigning is saying you win i give up. you can encourage them not to resign but can't force them.
And no, both players cannot resign. The first person who resigns loses.
"At the same time" is ambiguous
, not possible on chess.com anyway.
Resination is a chemical process that leaves a residue on the inside of your pipe. I guess it is possible for both players to do so at the same time. I would not try it in a state where it is not legal, unless you want to play a lot in prison.
I know a player may accept or refuse an offer of a draw, but can a player refuse a resignation? Can both players resign?