I use a router at home and it works fine. However, switching routers can cause troubles.
First off, what Teone says is a good idea. To see if it really is the router, plug it out and try to see if the problem is goes away. If it does then you know it is a router issue. You should probably still use a router but maybe replace it with some other router or install a firewall on your computer. It is a bad idea to run a computer without firewall directly on the global net - that is like opening your door wide open and have signposts near your home with "burglar, thief? Come here"

However, for a short time just to see if you can connect to EQ it should be ok enough. Do not leave the machine on and connected to the global network without firewall though. Also, running firewall software on the computer that runs EQ might in some cases interfere with EQ itself - depending on the firewall so as such a router has an advantage over firewall on the PC in that respect.
First off, routers do various tricky things with network. Most people have some form of NAT (network address translation). The good side, you have IP address on your computer which is often 192.168.X.Y or 10.X.Y.Z or some such. The truth is that at one time 32 bit numbers was "more than enough" for all computers in the world but today it isn\t enough. One solution has been NAT where both you and I can have a computer with IP address 192.168.1.5 for example and if we were both to go public with that address it would cause havoc since someone sending to 192.168.1.5 wouldn\t know if it were to go to my computer or yours. Therefore you have a public address provided by the ISP (internet service provider) which is often dynamic - i.e. it can change from time to time, this public address is yours alone. However, if you have 2 or three units at home that have IP address they all share that same public address. The router handles this. In addition to IP address you also often use something called a port number to indicate which application you want to communicate with and if you have a service on one machine 192.168.1.5 that uses port 5780 and another on 192.168.1.7 that also uses the same port 5780, the router will make a connection for each so that seen from outside your network they both have the same IP address (the IP address of hte router, the one that is public) but different port numbers, so if one packet comes in on port 1234 it sends it to 192.168.1.5 and if it comes in on port 2345 it sends it to 192.168.1.7. So, the good news is that all the computers can be on the net even though the 32 bit IP address is really too small to hold all the computers in the world.
The bad news is that what the IP address is as seen from outside is not the same as seen from inside. Outside a program think your address is 213.89.217.69 and port 1234 while inside your home network it really is 192.168.1.7 and port 5780. This can cause problems, it also allows the router to prrovide a special security feature. For example if you send out a packet on a given IP address and port and destianation to an EQ server and port the router can make such a connection as provided earlier but he can say that it is ONLY that particular EQ server and port that is allowed to send data to this port and it will reject any others. This is called "symmetric NAT" in which you have a 1-1 mapping between who can talk to which IP address and port on your computer.
In general this works very bad for EQ as EQ switches IP address whenever you zone to a new zone or when it switches from patch server to login server and from login server to a world server such as Saryrn.
If you have so-called symmetric NAT and you have the ability to turn it off for selected IP addresses you must do that for all EQ ip addresses. Tech support can tell you which IP addresses are used by EQ servers and you should allow them to be able to be reached unhindered. As far as I know, EQ servers never connect to your client - it is always the client that connect to them so it should be safe enough. Just allow OUTGOING packets to open such connections and make sure those connections are not symmetric, i..e if you originally sent a packet to A.B.C.D on port P then E.F.G.H on port Q should also be able to send a packet back to you on that connection provided it is in the range of the EQ servers.
If you have assymetric NAT this should be no problem, just make sure you allow your own machine to connect to any EQ server without being blocked. Firewalls must be configured so that EQ servers are acceptable targets for outgoing packets and also for incoming packets provided an outgoing packet has gone out first on the same source IP port. I.e. if you sent a packet from 192.168.1.5 (if that is your IP address) and port 5673 out to some EQ server and port then any EQ server (same or other) on any port should be allowed to send packets back to that same port.
Configuring routers can be a problem - especially if you are not an expert. See if you can find a local net guru to help you out but make sure it is someone you trust and that they do not sneak in any backdoor for themselves while they do it

If you do not trust anyone then perhaps asking the store where you bought the router should work, they should have people there who can help you out.