I did live dead studies for my hydrogel and i am using calcein AM and ethidium homodimer and few of my images show that all the cells are stained with both dye giving both green and red fluorescence from same cell
You may be keeping your cells with ethidium bromide for too long. If you keep live cells long enough with EtBr they will get stained. Also the cells may have started as live cell but died during the incubation - another reason to work promptly.
One thing you may want to rule out is auto-fluorescence. When cells die, they fluorescence in multiple channels, including the green and red channels you are using. Auto-fluorescence through the green channel is typically yellowish green, as opposed to the "nice" green you see with GFP and FITC (a control would help if available). I would recommend comparing your stained cells to cells stained with either dye alone and unstained cells. Hope this helps!