I have chained a shiny jigglypuff, gothorita, zoroark, pikachu, croagunk, and skrorupi. I can confirm that finding a shiny does not break your chain. You can go to the pokelab later and examine the machine on the second floor and it will tell you your top 3 records as well as your most recent chain. You should be at 41. Unfortunately, I have not been able to catch more than one shiny per chain and have not seen anyone report that they have gotten more than one in their chain. This leads me to believe that you can only get one shiny per chain.
A bummer, I know, but the good news is that your shiny is guaranteed to have 2 max IV stats. The bad news is that you only have one shot to get the right nature with your synchronizer.
I'm not looking for help, I'm complaining. This is GameFAQs, isn't it? ~ Shad0wflare