WAR ratings for Moreland, +.9, and JBJ, -.4, right now give Moreland the decisive edge. I actually think the gap is larger because Moreland's OPS is almost double JBJ's.
I also think JBJ doesn't make nearly the difference on defense that people usually give him credit for. I agree he's good, maybe gold glove good, but I also think 80% of defense is the guy on the mound, not the eight position players on the field with him.
There are actually 9 players on defense, including the pitcher. If you divide the defensive pie by 9, you get 2% and change for each. If you divide the 9 offensive players, including the DH, into 100%, the average contribution is 10% and change. That's a five to one ratio, hitting over defense.
You can make it closer by saying the pitcher only counts 60%, which is ridiculous, and by giving the centerfielder a higher than average responsibility. But I think the best you can justify is saying offense is twice as important as defense for any player with the possible exception of the catcher. I separate catchers out because they are so important on defense and because so few players want to be catchers or are able to become catchers.
JBJ has stayed in the lineup for big chunks of this season for several reasons: great glove, great pitching for the first 20 games or so, great hitting by the team, and a great won-lost record. Now that the Yankees have caught up and there are questions about the pitching, hitting has become that much more important, and JBJ's very weak bat is hurting the offense, especially when both catchers are not hitting well either. Oh, and that .510 of his is artificially high. It was .650 in April and is .212 (OPS, not OBP) in May.
Overall, however, I think the Sox want to keep him awhile because it's possible his hitting will come around as it has done in the past. Plus he will remain a world class defensive replacement if nothing else.