I doubt that thier immigration status will have any effect on the eviction proceedings. (I have never had to evict illegal aliens, since I ask all applicants if they have U.S. citizenship or a visa/green card to be here.) Immigration will do nothing unless, and until, they are convicted of a crime. (That's why there are millions of illegal immigrants here.) No you can't discriminate on national origin, but some areas of the country are now actually fining LLs who rent to illegal aliens. So if you ask it of all applicants, it is fair in order to ascertain their legal status.
What is the reason the court keeps adjourning? Have they given you a reason? If this continues, you may need to hire a flat fee eviction attorney so the court will quit jerking you around. Courts tend to look at attorneys different than LLs. If they truely are illegal, you may have trouble getting rid of them. They have nothing to lose, and may just disappear the night before they are to be set out. Good luck. Sorry, I don't have more to offer you.