If they don't pay within three days, we contact our attorney to file for eviction and unlawful detainer with the court.

The Case is now being handle by my lawyer who is recommending to file an unlawful detainer, while the squatters lawyer is arguing that his defendant has rights on the property like compensation, or plain rights to the propertyWhat are your thoughts?

In California, if a tenant gives notice that they are moving and then doesn't move out, you can immediately file an unlawful detainer action without giving any further notice on your part.

Screen your tenants thoroughly (income verification, past landlord check, unlawful detainer check, criminal background check, credit check).  

In Arizona, an eviction lawsuit is also called a special detainer action.

I also tell them that is what I will be doing and make them aware that unlawful detainer filings will come up when they seek to rent elsewhere.

If evidence was found an examiner can detain the violator, bring an arrest and prosecution.

Here's what I do - whenever I serve any type of notice (i.e. pay or quit, unlawful detainer, etc), I list every person named on the lease AND add something to the effect of "AND ALL OTHERS IN POSSESSION". 

Some say good-riddance when they are gone, others (Including me) think they should be held accountable for their actions and take appropriate steps to pursue them legally as much as feasible.Whenever I have a rotten-egg, I make sure to do all the necessary steps to remove them legally, such as unlawful detainer, possession order,  with filing an eviction and performing it as a mandatory step.

Smart money: hire a local law firm in DFW area with practice specialty in landlord/tenant law, specifically unlawful detainers.