if I cross the boarder illegally into Mexico can I then board a flight to South America without any problems?

No, you will have issues because you don't have your exit immigration papers with you. When you enter Mexico, the form you fill out has 2 parts, the officials keep the top part and gives you back the bottom part which you have to hold on to for when you leave the country.

i dont think so,you might have problems

Hell no ! Very stupid move you will be manhandled like a slave because Mexican authorities know you have no rights in there country since you are not complying with there laws . No one is going to help you 

Sure why not .......