My friend came into work sick and was throwing up every 5 minutes. Her boss refused to let her leave and put her on drive thru with a trash can next to her. Is it illegal to keep a worker on the clock if they're physically ill?

First time sick? Why come to work ilI? Thowing up? Perhaps she should have called or visited her doctor for a excuses sick day off? Couple days? 

Illegal, she'll have to check with HR on sick policy. 

I knew a woman who was always sick, she could make herself cry, throw up and fall out at will. Last "sickness" episode employer told her she could go home and file for unemployment or use the trash can. She throw up all day.  2 days later they fired her.

That being said it depends on your company's sick policy.

Regardless of the job, if she is handling food and throwing up at the same time that is grossly unsanitary and should be reported to the local health department.

If she is in the US, it probably depends on what state she is in.  And if there is a union, altho being a fast food establishment there probably is no union (or HR dept).  If she is in a progressive state, she might have some recourse.  If she is in a "right-to-work" state, she has no recourse.  "Right to work" means the right to shut up and do what the boss tells you.  Of course the health dept might be interested, and the customers except if either finds out she will be out of a job. 

Yes, it is.  Holding someone against their will is against the law and so is holding someone against their will for fear of losing their job, is very much against the law.  There are labour laws that will dictate what to do and what to expect in a situation like that.  Laws and protocol differ from place to place so you would have to look it up to see what pertains to your area. However, "Right to work" states may offer little to no protection from being fired.

If this person had a contageous disease like the flu (for example) the employer may also be in trouble for causing a health hazard to other employees.  I would definitely look into this.