You should NOT need a cross over cable from a PC/Mac to a switch. A Cross over cable is use to connect 2 like devices like from Switch to switch OR Mac/PC TO Mac/PC. UNLESS your switch has a uplink port (used to connect two Switches routers or a Modem to the Router Essentially it is a port that allows you to connect 2 Like Devices without a Crossover cable. IF your Mac is connecting to a router with a crossover cable to the normal port it SHOULD NOT work as the send wires of your Mac will be connecting to the send of the Router and Vise versa. If it is working then it is really NOT a cross over cable OR you are connecting to a uplink port (the port is switching the wires back so it is working like a normal cable anyway.
Also your home is wired (with Jacks you then connect patch cables to) check the Jacks are they using T568A or T568B it does not matter ether will work as long as ALL the cable ends are the same. If you have one end of a cable wired to 568A and the other end to 568B It won't work BUT it could work at 10baseT as only 2 pair are used. Also if you use a cable wired to 568A (at both ends) and connect to a 568B jack or vise versa it will still work as as long as the wires are the same at both ends of the same cable. But it is Best to just use one or the other on ALL cables to avoid confusion.
Also the real issue is when It is Nether 568A or 568B in that it is wired incorrectly by switching 2 wires at one end (if it s a pre wired commercial cable it is most likely wired straight though at 568B
Also Cat6 is recommended for 1Gb BUT 5e will be fine for SHORT runs as long as all 4 wires are connecting and the cable is wired correctly Meaning ALL 4 wires are on the same pin at each end NO Cross over or Flopped (Flopped is when pin 1 at one end is wired to pin 8 pin 2 to 7 and so on at the other this is also known as a cisco Serial cable.) (some people confuse a flopped cable for a crossover cable they are not the same)
MOST ethernet in the US is wired at 568B and as long as both ends match and the wire can handle 250MHz (which is Cat 6 standard) A Cat5e will for a short patch as the loss over a short patch is not as great as it is in longer run BUT if you really need 1Gb speeds (High band width data like Video or Linked processing (3D rendering over network) Cat 6 is really recommend.
That said IF one PC or Mac works fine ON THE SAME CABLE and ONLY the Mac Mini is having a problem it could be the Negotiation between the Network card used in the Mini and the router/switch
I would really like a update from the Mac Mini users as that was the original thread as those users seemed to have their network working FINE except for the Mac Mini so blaming the cable or the Switch (unless it is a older switch) seems like a deflection of what could be a real issue.
Also this could be a much bigger issue then is reported to Apple as I would bet most Mac Mini users use Wifi and only use the network connection to connect to the internet and unless they really have a VERY fast Service plan the 100baseT would be faster then their internet connection anyway. So even if they hard wired direct to their modem they would get little benefit or notice any loss if 1Gb ethernet did not work. Only users that need to transfer large amounts of data (over a short time) within their home network would even notice (or if they checked)
And if you have only ONE Mac and your network iS ONLY primarily only to connect to the internet and maybe a network printer then Again 100baseT is plenty fast enough to handle even better cable modem speeds And most print jobs will start to print almost instantly at 100baseT so again most people would never even notice. (Comcast or other cable intenet basic package is around 30Mb down 5Mb with there best only going to around 100Mb down so 1Gb is over kill to for just internet)