Should I wear a surgical mask while soldering circuit boards?

YES if you are soldering as job (8 - 12 shifts) the filtering of flux smoke would be prudent.

Health and Safety Regulations,in Western Societies, tend to make it mandatory to wear a Mask, Rated for the level of 'toxic air borne threat', in which you are working.

Short term one-off home job should be OK but still wise to wear a mask, cheap (disposable mask would do) but also ensure you have plenty of fresh air ventilation.

Hope the project goes well and try and find a better job if you are a cog in the production line.

A surgical mask won't help at all, just work in a well ventilated area and try not to inhale any more of the fumes than absolutely necessary to get the job done.

If you're working in a decent company, they should already have adequate ventilation installed to keep you safe.

You can if you want to but there is no medical reason to do so. Just don't sit there snorting the fumes from the flux.