The 21-year-old was set to re-sign for the Daggers after making 36 appearances last season as they reached the National League play-offs.
Staunton, who can play as a centre-back, right-back or midfielder, joined Dagenham in June 2016 but turned down a new offer when told there would be no guarantees over his playing time.
"Josh Staunton leaving was a strange one for me as we had agreed everything with Josh and we had no problems," Still explained to the club's website.
"When Josh came back from holiday he had a conversation with me about whether I felt he was going to be a starter this year but with all players I can’t give that assurance to anybody.
"How do I know how anybody is going to perform pre-season? I said that to him and that I couldn’t give him any guarantee, it’s whoever is the best in that position will start, he was looking for more assurance than what I could give and he decided he would go somewhere that might give him that, but I can’t see clubs giving that to him as they do not know how a player is going to play in pre-season.
"I could have lied to him and said Josh you will play but that’s not the right thing to do.
"We are quite comfortable, though, as we have brought in what I believe to be very good players, as well bringing in players with potential. I believe the entire close season for us has been very good."