Advancement (through quest, or otherwise) needs to be tied into some form of achievement. My characters are all Republic, so believe me, I feel your pain. However, simply awarding gear for participation would lead to an epidemic of AFK'ing, or of players doing the minimum possible to get their stuff.
The Medal system is so flawed and skewed to Tanks/DPS over Healers that it would not serve in its current form. MVP wouldn't work either, as most players either don't vote or simply vote for the player with the biggest Damage or biggest Heals value.
So, what else is there other than winning?
All of these suggestions are ignoring the TRUE PROBLEM, which is not how daily quests are completed, but rather how Warzone teams are formed. If Bioware can, as they are suggesting, drastically improve how players are matched against one another by a ranking system and/or by going cross server for a bigger pool of players, then chances are that we'll begin seeing more equitable Warzone matches and you'll stop losing six in a row.