Most likely not. If the top of the tooth break, the roots are still in the jawbone and can become infected and cause an abscess. I would suggest trying to save some money to get the tooth removed to prevent a problem. Even if you decide to hold on to the tooth until it hurts, which some people decide to do, it would be a good idea to have the money readily available for the necessary treatment. But the longer you wait, the more damage can occur and an abscess can spread rapidly.
the tooth that is breaking has had a root canal done to it. If the tooth does break apart like an egg won't the rest of the tooth just fall out?
Well if the tooth is breaking apart, it most likely has a cavity in it and its making the tooth weaker. That's usually due to the bacteria penetrating the enamel (the hard outside of the tooth) and getting into the softer layer underneath. Once its in the softer spot, it'll continue to get larger underneath the tooth. When the layer underneath is too weak, the tooth will crack and break apart like an egg.
how can you tell if the tooth is infected?
The ER probably won't be able to remove the tooth although they can prescribe antibiotics if there's an abscess or the infection spreads or pain medication if it starts to hurt. Most hospitals don't have a dentist or oral surgeon as part of their regular emergency staff.
there is no free dental clinic around here. I can't pay the amount of the reduced price they have told me to pay either. So the tooth is just going to have to fall apart on it's own. If it is bad enough I'd probably have to go to the ER.
You may need to contact your local dental society to find a free clinic that may be able to take care of it for you as a reduced or free cost. Til then, you should try to avoid eating on that side but there's no way to repair the tooth permanently without a dentist.