I hate to say this, but whoever you dealt with in tech support is - ahh - clueless.
First - if we're discussing the link lights as in your first post, then, as you mention in your second post, ip addressing is of no relevance whatsoever.
The DNS-323 does not need a "network that is managed by a router", in fact, as I believe you already know having connected it directly to your PC, it doesn't even need a switch - you can connect it directly to a PC, assign the PC a static ip address in the 192.168.0.x network range (anything other than 192.168.0.32), allow the DNS-323 to default to 192.168.0.32 and use it like that.
I see no reason why you should have problems using a Netgear GS116 switch - although an auto negotiation problem is one possible cause of your problem. My DNS-323 is currently connected to a Netgear GS108T, and prior to that a Netgear FS728TS and I believe a Netgear FS726T - the basic difference between my Netgear switches and yours is that mine are SmartSwitches and have some degree of management capabilities allowing me to fiddle with the speed & duplex settings - IF - I need to, something that I have never done with these.
For the record - I have a Netgear router, but, my IP management is handled by the DHCP service on my Windows 2008 server. My DNS-323 does have a static ip address, as I consider static addresses on servers to be a mandatory requirement. If it were not my network, I would rephrase that to "for servers, static ip addressing is strongly suggested", but since it's my network, my rules apply, and it is mandatory.
Back to the problem at hand - try a different cable, if you have not already done so - even if the same cable works with your PC - gigabit networks can be finicky where cables are concerned, especially when it comes to length and the "pairing" (which wires connect to which pins in the connector) - there is actually only one correct way to wire the cable, but, the slower networks are more tolerant of miswired cables. If necessary, move the DNS-323 so it's right next to the switch and use a short "known good" patch cord.
You can also try locking the speed of the DNS-323 down to 100 and see what happens.