Should I use 'them' or 'their' ?

I would like to know please, which of these two sentences is correct and why?

Are both acceptable? Also if possible, I would like to be told what part of speech the 'them' is and the 'their' is in each case.

Option 1

The home stay couple I stayed with in America was wonderful and I loved them, but they were both very big and I worry about them having a heart attack when they get older.

Option 2

The home stay couple I stayed with in America was wonderful and I loved them, but they were both very big and I worry about their having a heart attack when they get older.

Thank you,

Tony