If you data acquisition board has multiple analog input channels, then you can acquire all of those channels on a single computer. The BNC box should give you access to multiple channels. You simply connect the multiple signals up and then tell your software to acquire from a collection of channels. If you are using LabVIEW, simply specify which channels you want to acquire from in the channel list (e.g. 0,1,2,3). I hope this information helps.
