No, the router doesn't matter, almost all routers allow any outgoing.
Certain router software (Astaro) is overly paranoid and you have to poke holes in the firewall in both directions, but I don't know of any consumer router that does this by default.
The simplest solution is to use the cox SMTP server and whatever cox email ID you have as your From address. You can still use any To address you wish.
I'm not in the US or near anything Cox-ish to know what auth you might need or not, maybe a snoop on the cox web page would yield a howto for email setup, that'll let you know what they allow.