How do I rewire this electrical panel?
The neutrals are hot until they're connected to the neutral bus whenever the load is connected (switched on or whatever). You have a very dangerous situation there whenever they're disconnected. You should never disconnect neutrals or grounds unless the associated hot is disconnected or the breaker switched off.

More info and explanation

In a main panel, ground and neutral are "bonded", and they share one or more bus bars. The neutrals and grounds for a particular branch circuit should never be installed together under a single screw. It is a code requirement that each neutral be terminated one per hole with no other wires. This prevents inadvertent and dangerous detachment of additional neutrals when working with a single screw.

UPDATE: Here's a good discussion on 408.21, the section that covers this issue. My interpretation is that neutrals must be individually secured, but grounds can be doubled (or even tripled) when the hardware allows. Confusion often stems from the terms "grounded" and "grounding". The former refers to grounded neutrals, and the latter to (typically) bare ground conductors.