Recent immigrants? I assume you weren't trying to be racist. Hell half the recent immigrants I know and I do know quite a few speak better english than half the native born Americans I know. If "We all got our problems" is grammatically correct and our is the possessive form of we and they is the possessive form of their then where's the problem?
Go to this url -
http://dictionary.reference.com/search?q=they - and read the usage note for "they" and you will see that all though technically "Everyone has their problems" is wrong it is still accepted by over half the people (not hillbillies) tested. Which is basically what I said in my comment. Most of the people who read the first version of the poem weren't confused by the line, I'm sure. I also stated that it was inappropriate to use such casual grammar in this type of poem. A hillbilly would say "Everyone gots them some problems". I know plenty of hillbillies too. A male chauvinist would say "Everyone has HIS own problems". I never said "Everyone has their own problems" would be a perfectly fine. Just that it would be clearer than just "their" which could be interpreted as each other's problems. I also never said to use it in this poem. In fact I advised against it. My point was to assure the poet that grammar doesn't always have to be perfect in poetry if you have a reason to use common less perfect language. I didn't mean to step all over your comment I just didn't want the author to be permanently scared of using slang. You can be intimidating sometimes Zodiac. Verily I agree with your first comment and I still think your smarter than me.