If you intend to use the fireplace, you will need something better than cement board and tile. You will need high heat cement and firebrick. the high heat cement is actually ceramic based, not cement based, and will get stronger with use, not weaker.

Cement board will work fine for a hearth. However, you will probably need two layers since most codes require at least an inch of non-combusitble material on the hearth (check your local code). It's highly unlikely that the hearth has any structural connection to the fire box. However, there could be steel that is connected. You may want to use a concrete saw to make a clean cut at the junction of the hearth and firebox. As an alternative, you might want to consider putting down cement board over your existing concrete hearth and tiling it. That way you save a lot of demolition, the hearth will only be raised slightly (1/2"?) and the tiles shouldn't come lose since they will be on a solid sheet of cement board.