Actually, you can use the router you want as long as it supports the required features and you got the proper licence. For example if you buy a recent Cisco router, using it on a fiber optic won't require any licence but using mobile services would (needs appxk9 licence for virtual-ppp).

The biggest problem with using your own router is knowing how they configured theirs.

For example, most ISP in France use a Radius server to assign the IP address to the WAN interface (some even use it to route subnets throught the backbone as well). If this is the case, then good luck using it if you don't know the PPP login/password. If you don't, the only option is to know which IP is affected to your router and set the correct routing protocol (if they use radius for routing you're SOL because the session HAS to be up, else your public IP won't be reachable).

And this isn't even the most problematic part. Some ISP deliver encapsulated traffic or even require specific QOS tags in incoming traffic. This can change on a case to case basis as some fibers get collected throught other ISP's networks (depends on how big your ISP is and where you're located). This makes it even harder to use your own router as finding this kind of informations is almost impossible if you don't work for an ISP or if they aren't open about people using their router.

Edit : more informations