Can a soldier who is assigned to CQ be taken off for half a day to perform another detail and then put back on CQ for the remainder?

I've never heard of a problem with doing this. Regardless of where the soldier spends that time, they are going to be on duty for 24hours anyways. Frankly, I've had soldiers who would much rather be somewhat busy doing something for the time to go by instead of sitting at a desk and counting the holes in the ceiling. This also depends on the duties that are placed on your CQ. Is it the NCO or the runner that is leaving? Are there usually circumstances that require one of those on CQ to leave for a period of time (not counting latrine breaks)?
I would actually like more information on why this question may be necessary because it just seems like one of those where someone is just trying to argue over regulation semantics or something in order to get out of doing something.

I would say after looking at 220-45, AR 600-20 and FM 22-6, which are listed as the supporting required manuals in the 2012 edition that the answer would be yes. Because CQ is essentially interior guard and there is a discretionary statement in the FM that allows the commander of the relief to direct the soldiers it also discusses special orders and SOPs.
Personally I would avoid doing it , and don't remember having done it. Our CQ had security checks and barracks visibility walks that were required and you needed two troops. unless something absolutely went to heck because you could end up in a situation where the post went unmanned. But it's not cut and dry.
I think you would end up in a "I told you so" situation because it's a judgment call and isn't spelled out, none of us know your units task load or manning situation well enough to poke your leaders too hard..

I've seen it happen. Now whether or not its permissible within regs might be another story; wish I could say I knew. As inconvenient as it may be, I don't know if losing your guy for half a day is worth fighting the CoC over it. Good luck.