LMZ34002: Can you help recommend the suitable Part number?

Hi,

You can use LMZ34002 to generate the negative voltage, for the positive one, you may need a buck-boost converter. I am adding buck-boost team in your post.

Elsa,

unfortunately, i am not aware there is one chip solution for two output voltage rail with >1.2A current capability. 

the TPS55288 or a boost device in Sepic circuit could be possible in positive 5V rail, but can't support -5V rail. maybe you need another buck from 5V to -5V