And any application I have ever seen with a solar cell and a battery, they are hooked up in parallel so that either device can power the device. An example of this is a dual power calculator where the solar cell will make sure your calculator works in the event the batteries die (assuming there is enough light). Normally the batteries here are not rechargeable.
On larger applications solar cells can be used to charge batteries, and then the DC current can be put through an inverter to provide AC power.
For a paintball gun it should be a DC circuit, as I can't think of a board I have encountered with an inverter on it,.... but I could be wrong as I haven't seen circuit diagrams for too many boards, but if its DC or AC it won't matter, as the power source for an of them is a battery.
I am not overly familiar with solar cells so I don't know what to tell you for size wise what you are looking for. If you can find a part number on them though you should be able to find a datasheet on it. You will need to make sure you can get enough to provide the voltage and current for the gun to operate. Voltage probably won't be too much of an issue,... if you can't get close enough without going over you can always build a resistive voltage divider, or drop it through some LEDs to a power indication as well as knocking a little voltage off each time. your bigger problem would be current,... and I have no idea what the typical current draw is on paintball guns. It wouldn't be too hard to find out though, you could hook up a DMM in series with the battery,but you might need a little butchering of the harness to do that. Once you know what the maximum current you will need is (go through the guns different modes, try firing etc.) Once you know this, and the max current you solar cells can put out, you can figure out how many sets of cells you need to hookup in parallel to get you current.
Finally I would recommend keeping the battery in the gun and hooking it up in parallel to the solar cells, in case you are playing in low light conditions. If you do this put a diode in series with the battery, and another in series with the solar cell to ensure current can only flow in one direction. This will prevent you from trying to charge the battery (some batteries have very specific charging needs so I would stick to using the proper charger for them, unless you are sure on what you are doing for your specific battery, and its also possible to over charge a battery and permanently damage it).
As a side note, you said the solar cells are broken, or was this the devices that they were used in that are broken? If its the cells themselves that are broken (as in broken in half or some other way) the data sheets I mentioned earlier probably won't do you a lot of good.
