Is this "chain migration" they keep talking about only for the Dreamers - i.e. they don't want legalized dreamers to bring in all their parents, etc. or is it for ALL immigrants who are currently legal and/or citizens and they're trying to stop all that?

In my opinion, if they want to stop chain migration for dreamers until we are citizens, then sure, have at it. Eventually we WILL be citizens. If they're trying to prevent citizens from sponsoring their family members then that is a scary poison pill that will affect everybody long after the Dreamer debate is gone.