Can a school district monitor your internet use on a private network off school campus?

I live in Long Island, New York, and my school district has put in place a Wi-Fi network for student use, which would have the expected Iprism firewall. The only thing with this is that in orer to use this service, you must first be redrected to another company's page where you have to "register" your device. Registering your device entails downloading something that will configure and alter your phone (I have an iPhone) , tablet, iPod, and computer's (MacBook Pro) settings, in order to use the network. I'm extremely uncomfortable with this because it involves installing something to use a service which alters my settings, and by law monitors my use on the network, as well as IP and other stuff. But the thing that really bothers me is what if the thing i need to install not only monitors me on the district network, but also on my home network or other network and keeps tabs and sends the information to school officials and/or authorities. If this actually happens, would this violate my right to privacy, and can the district as well as company that the district hired to engineer this program be accountable and prosecutable under the law? (Also, keep in mind, I'm 17 at the time of writing, but in november (2 months from now) i will be 18 (also a senior in high school.

Added (1). Let me just add a bit of clarification. As I stated before, I'm well aware that by law they have to monitor my use of their internet network. I have that down, I don't have a problem with that. What i'm asking is what if the software they are asking us to download monitors our use of the internet on networks off campus, public like starbucks, or private like at home. Is it in fact an invasion of my privacy rights, or are they allowed to do so?

They can't monitor what you do on your home network. In fact, that would probably be illegal.
They have all the rights to monitor what any device does on their network, for legal reasons, but they have no rights to monitor what you do on your home network (or a public network like Starbucks)

Check with the school what the software actually monitors, and if they say anything about home or public network, ask them to remove it because it violates your privacy rights…

They can't monitor what you do away from their network unless they have software installed on the laptop which records such things.

Yes it's school property
so they can say they must monitor what happens on it

I would guess that the reasoning behind being able to monitor your online activity when using their network, is to prevent someone from getting a virus from some web site, and letting it run loose when they connected to the school's network. From your explanation, you don't have to use the school's Wi-Fi service if you have your own.

You said, "by law they have to monitor my use of their internet network", but I've been unable to locate the requirement, so I'm beginning to doubt that it is so.

The school district is required to document and make publicly available the reasons for any monitoring, what overall data and what personally identifiable data is collected, how the data is used, how personally identifiable data is protected, and a complete list of the parties with whom it is shared.

It is very important to have a hardcopy of the actual document, and not to rely on hearsay that can later be denied, or on an electronic document that can be arbitrarily altered (it is trivially easy to do this without changing the timestamp).

Monitoring software that continues to track your activity away from the school's network is a violation of federal You also can't be coerced into accepting the illegal monitoring, for example by the threat of being barred from the (It is generally illegal to consent to an illegal activity.)