I have no idea where the half double crochet stitch is supposed to go in at the end of the row and how you're supposed to DC around it.
I'll try and help with both parts of your question - where is the half double crochet stitch supposed to go in - this is what the pattern says: join with hdc in 3rd ch of beg ch 3 to make corner
So, when you get around the square to this point, make the half double crochet in the top of the beginning chain 3 from the same round.
and this is part of the disucssion we had earlier on how you are supposed to double crochet around it (on pages 9 and 10):
at the end of the first round you made 1 hdc into the 3rd of beginning 3 ch. That hdc formed a loop, which is the corner of the square. You insert your hook into that loop. Hope this helps.
Be sure to insert the hook INTO the loop!
If this doesn't help, ask again and maybe some one else can explain it more clearly!