[Pov]

Assuming neither queen was a club your friend was 31% to win pre-flop. After that favorable flop he was probably closer to 45%.

Of course the only stat that matters is the one from when he made the decision. His random hand against 9 other random hands wins about 10% but since so many people will fold it probably wins very frequently - but just a teeny tiny pot like you point out. For fun though, let's try to put numbers to it.

Let's say there are 9 other players and they will only call with AA, KK, QQ, JJ and AK. That's about 3% of hands. So (simplifying) we'll say he wins uncontested 73% of the time. If he does in fact get called he's going to win approximately 25% of the time against this range of hands. Just to simplify further we'll assume that everyone has him covered so he either doubles up or busts. That would give this play an EV of:

(.73 * $3) + (.27 * ((.25 * $283) + (.75 * -$280))) ~= -$35.41

If he's not actually covered by most of the players then it's not quite as bad, but obviously still very very -EV.

Math corrections are welcome.