Can Steam players and Origin players play together?

If it is possible for Steam Players and Origin Players to join, I would like to know if there is any way to identify who's using what.

I can't use the Steam Client, and any IP Addresses associated with Steam tend to crash my games (assuming they even get that far). I would need to be able to filter out Origin Client Lobbies and prevent Steam Users from joining my games, as to not crash my Router.

It's been a thing in my case for years, and nothing will fix it (requires calling the ISP and going through a lot of menus and so on over a phone for hours). Steam got Black-Listed on my Router years ago, due to some Hacking incident involving the Steam Client and another User I think. Ever since, people using Steam on their systems have found themselves knocking my Router offline every time they join me Online (hard enough getting a stable game of Battlefield 4 with other users have both Steam and Origin active at the same time).