No, they do NOT have the right to claim asylum if they are here illegally.  Why not?  The relevant law (passed by Congress and signed into law by the president, as the constitution requires) is https://www.law.cornell.edu/uscode/text/8/1158 and states quite clearly that they do have that right. No judge, no matter who appointed them, is going to rule that the executive branch can just ignore this law...  and that’s why the Supreme Court did not overrule either the “Obama judge” or the other “judge in San Francisco” who was appointed by Bush and ruled the same way.