I have lived in an apartment owned by my sons grandparents for 15 years. I had an oral agreement that we did not have to pay rent and we'd always have a place to stay. This agreement is supposed to stay in effect until I graduate from school and get a good job. I graduate in two weeks, and received a letter stating that the apartmtent has been turned over to my sons uncle and he wants 100 dollars a week. He has not spoken to me and there is nothing in writing agreeing to the rent. I would not have any problem paying the money if he could have waited a bit longer for me to graduate and start working. I am in a financial hardship right now having paid my school 2,000 dollars to take my last class, I had an accident where my car has been totalled and many other bills problems etc. that I am behind on being in school full time without work.
I have been served an eviction notice from the court and am concerned about what my son and I are going to do. We do not have money to pay them and do not know if there are any laws that would protect us? At least until I am able to make some money working.
Please help!

You mentioned "squatter's rights" in the subject, so I will address that - in a nutshell, there are no "squatter's rights" except under some very specific and unusual situations. Nothing in your case leads me to think that you have any rights beyond those of a normal tenant. Given that there is no written lease agreement you are considered to be a month-to-month tenant, and as such the landlord can require that you move out with 30 days notice. After the 30 days if you haven't moved he can start the court process for eviction, and ultimately get the sheriff involved.

In Florida a landlord in a month-to-month tenancy has to give a minimum of 15 days notice.

... I ... recieved a letter stating that the apartmtent has been turned over to my sons uncle and he wants 100 dollars a week. ...
I have been served an eviction notice from the court ...

You were sent the letter and then received a notice from the court? I don't see when, between these two notices, you said you were given a notice to quit. As Ebaines indicated, you must be given a 15-day notice before a lawsuit can be filed. Did you get one of those too?

Several things, you needed to receive a notice that the rent was going to change, from 0 to 100 dollars. Then after you did not pay, they would give you a pay or quit notice, and then if you did not pay, you get evicted though court.

You need to work something out with the new owner, I would be at their house in the morning and see what you can work out, ( in writing but appear in court with agreement to be sure it is covered)

I'm not so sure of the timeline here. I would say you had to be given 15 days notice that rent would now be charged as of x date. If you didn't pay by the due date, you could then be given a 3 day pay or quit notice. If you didn't pay you should have gotten a notice for a hearing. A the hearing, you would have been given notice to vacate.

so what has actually happened here, exactly.