Can I use 14/3 wire to connect a new switch on a branch in a 4-way circuit?
You'll need something a bit fancier than a standard switch to bail you out of this mess

With standard four-way switches, a mid-loop spur like the one between boxes 2 and 3 would require a /4 cable instead of the /3 you've run.  However, a more sophisticated system like the Lutron Maestro can handle this without a hitch: simply replace the switch in box 1 with a MA-S8AM and the rest of the switches with MA-AS'es.  These can work with only a line, a load, and a single interconnect wire connecting the accessory switches together, which allows for the mid-loop spur you have to be run with the cable already in the wall.