Well without throwing gasoline on this small fire, I will give you a list of very, very possible culprits:
1) yes indeed your backer board is way to thin. 2) did you over screw, not nail this to the subfloor? You need a screw inserted every 8 to 10 inches along the joists. 3) See, if number one and two were this way, then as you walk the 1/4 inch backer board is flexing and there for cracking the grout. 4) This will not stop cracking even as you re grout.
Here is what I would do.
this is an experiment but a try.
I would remove carefully every third or forth tile and screw in some 2 1/2 screws into the backer board then into the subfloor. What you want to do is to properly secure the backer board to the sub floor to eliminate flexing.
Then re mastic and grout these areas. you need to do this to a great deal of tiles.
If this does not work, I would then go to plan B and rip up everything and even the backerboard and install much thicker board and properly screw it to the sub floor. Then start over.