It's not the WiFi router that matters, but Virgin's V6 & TV360 boxes need VM broadband to work.
The internet-based services (OnDemand, streaming & EPG updates) won't work over a non-VM connection.
If I switch my internet provider and keep my TV package with virgin, can i use my v6 boxes with my new wifi?
If you cancel your VM Broadband it will also cancel your TV services. Although some services come through the VM internet hub, they are exclusive to VMs network, so other providers hubs cannot access them from outside.
The main issue is that the TV boxes will get deactivated and need to "phone home" at intervals to say "Hi, still in use" in order to prevent deactivation, and phoning home can only be done by using the hub. This is why all boxes after the TiVO, currently the V6 and 360, must be sold as part of a package with broadband provided by one of VM's hubs, and VM won't permit you to remove only broadband.
If you're not too bothered about recording programmes, then VM's broadband plus a Stream box could work well, otherwise go for BT TV (requires BT broadband) or Sky (works with any ISP).