I think I know what the problem is, and Bioware needs to read this effing ASAP:
The problem seems to be with the possibility that, when you choose a daily, that is not the same as choosing the mission for the first time. So there'll be a phasing conflict.
Same problem, if the above is not the issue, but different parameters: It may lie with the possibility that ALL players need to take the quest at the same time so that everyone participates in the GROUP DIALOGUE from the start.
In other words, if one player already has taken the Flashpoint mission and is ready to enter, then he needs to abandon the mission so that everyone can participate in the dialogue. I wasn't able to confirm this, because some people in my group refused to abandon their quests - or they did it but retook it outside of group dialgue.
This is a total mess. And some of those people even mentioned quitting SWTOR if the problem kept up.
Please FIX THIS, Bioware. Thanks.
I too have this problem. We tried what Zoroth suggested, all of us dropped the quest and started from the beginning together. We still could not enter the Flashpoint after that. I figured out I was the problem and could not even enter flashpoints by my self. So when someone has this issue, it may just be 1 person in the group thats having the problem.