When our SCE shuts off the power, will Cox Internet still work?

I'm not sure if your question is whether Cox has a backup power supply or not.  The Internet will still be available without local power, but you have to go through your ISP to get to it.  That means both you and Cox must have a power supply.  But, do you have anything mission critical that depends on the Internet?  If you do, then you need to invest in a generator to ensure you always have power.  If your electric company shuts off power to Cox, they're probably shutting off power to you too.  Even if the Internet is working and Cox is functioning properly, it won't matter to you if you don't have power.  It seems to me that not having power would be a bigger concern than the Internet not being available.  

yes, if you have a portable generator.