Despite the advice you were given, it's quite unlikely to be the switch. Most managed switches do have some filtering and routing capabilities, but they all default to being off. Unless you've explicitly enabled them, the switch shouldn't be filtering anything.
There are a few things I can think of to check (and I know from your thread that you checked many of these, but it never hurts to start from the beginning):
Make sure all the clients are on the same broadcast domain. In your case, that means all connected to the switch, and all on the same VLAN (if you are using VLANs). Particularly if you are using wireless, this might be the issue. Some router/modem combination devices do not pass broadcast traffic between the wired and wireless ports. Test with the computers all connected to the same switch by ethernet.
Make sure LAN sync isn't disabled in the dropbox options on each computer. It can disable itself under some circumstances.
Make sure no firewalls are blocking dropbox traffic. This may also include some antivirus programs that interfere with broadcast traffic
If all that checks out, breakout Wireshark on two machines that should be syncing with each other and make sure they see each other's broadcast traffic. Wireshark has decoders for dropbox traffic, so it will show up as "Dropbox LAN Sync Discovery Protocol". If you don't see any such traffic, or you do see the packets from computer A on computer B and vis versa, then it's not a network issue. If you see them being sent, but not received on the other PCs, something is blocking it.
