Do negative counters trigger Protean Hydra?
Yes, it is removed and you get the EOT trigger. Per Gatherer:


  If a -1/-1 counter is put on Protean Hydra while it has +1/+1 counters on it, that -1/-1 counter and a +1/+1 counter will be removed from it as a state-based action. This will cause Protean Hydra's last ability to trigger.