odd toileting behaviors

Question
We have a one-year-old jack russell/beagle mix we adopted from the humane society.  She seems to have had a lot of trauma in her life before us, but we aren't sure exactly to what extent this affects her behavior.  She is phobic about confined spaces (not able to be in a crate without completely decompensating) and is currently showing dominance behaviors that we're also trying to break (she just recently started trying to hump everyone in the family).  Our biggest problem is that she refuses to defecate while on a leash on a walk.  We have a doggie door that she has access to while at home during the day.  When she stays with others, she refuses to go outside for anything other than urinating.  She has created quite a mess at my parents' house, and I'm concerned if we can't get her to "go" on a leash she won't be welcome there any more.  I know she gets distracted while out seeing the world, but is there a way to know to do her business when she is supposed to?

Answer
She's most likely fearful of defecating in front of anyone because she was abused by someone due to the presence of feces.  This is not easily cured.  You need a very, very long training leash (or extending leash) to TAKE HER OUTDOORS and you need to know more or less when the dog needs to defecate by observing her habits in the home.  A doggie door won't do it.  As for her being welcome at your parents' home, there are (if you do your homework, inspect the premises, ask for references from people who use it) boarding facilities available.  Obviously a crate was abusively used (she was confined overlong, perhaps defecated in the crate and was beaten for it).  This is a trust issue.  It will take many, many weeks for this dog to lose her fear and begin to trust.  Feeding a high quality food (like Royal Canin) creates hard stool which does not make a mess.  Put the dog on an indoor tab (long lightweight leash) when you are at home so you have control over how far she can get from you.  She will show signals of having to defecate; there's only so long any living creature can contain the urge.  Take her out when you see an anal bulge (short haired dogs are easy to spot when feces is present in the rectum) and then have a pocketful of special treats (cheese) to give her after she eliminates outside.  Don't give up on this dog.  You're her last stop.