You have admitted trespassing on this public forum, which is not likely a good idea. Also, school property is not likely considered "private". You should get a juvenile criminal defense lawyer to help you defend your ticket. I recommend Kenneth Newell--he has an office in Lake Worth and handles these types of cases.
*Disclaimer: Visiting or sending communications through this website does not create an attorney-client relationship. The answer to this question is of a general nature. All representations by the attorney in this answer may be affected by and contingent upon additional facts and circumstances of the individual case. Please consult a reputable attorney to discuss a full and complete assessment of your case.
Your title asks about private property but then your question refers to being on school property. If this is a public school, then it would definitely be public property. Even if a private school, then it would probably fall under a local curfew ordinance. If you are under 17, then it is a juvenile offense. If you are 17 or over, then it is not a juvenile offense. The difference is basically academic in your case because you were probably issued a citation ('ticket') by a city and will be in municipal court, not the county juvenile court. Most city ordinances are online and you can look them up to see the one that you are accused of violating. Finally, as the first attorney noted, you should not post admissions about violations to the law on the internet.
Answering your question does not create an attorney-client relationship. Furthermore, you should not take the answer as legal advice or a complete answer to your question because the facts of your case can make a big difference and it is impossible to know the advice to give without a consultation. You should seek a consultation with an attorney immediately to protect your interests and rights.
Yes, they can do this.
As the other two attorneys wrote, you need to consult a criminal defense attorney. DO NOT post anything else concerning this on this website or through any social media site. The prosecutor and the police can use anything you post against you in court.